Numbers

1. (935421 x 625) = ?A. 542622125 B. 584632125C. 544638125 D. 584638125

Hide AnswerHere is the answer and explanation

Answer : Option D

Explanation :

935421×625=935421×54=935421×(102)4=935421×1000016=584638125

2. Which of the following is a prime number ?A. 9 B. 8C. 4 D. 2


Answer : Option D

Explanation :

2 is a prime number

A prime number is a natural number greater than 1 which has no positive divisors

other than 1 and itself.

Hence the primer numbers are 2,3,5,7,11,13,17,...

3. What is the largest 4 digit number exactly divisible by 88?A. 9944 B. 9999C. 9988 D. 9900


Answer : Option A

Explanation :

Largest 4 digit number = 9999

9999 ÷ 88 = 113, remainder = 55

Hence largest 4 digit number exactly divisible by 88 = 9999 - 55 = 9944

4. {(481 + 426)2 - 4 x 481 x 426} = ?A. 3025 B. 4200C. 3060 D. 3210


Answer : Option A

Explanation :

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

Here, the given statement is like (a + b)2 - 4ab where a= 481 and b = 426

(a + b)2 - 4ab = (a2 + 2ab + b2) - 4ab = a2 - 2ab + b2 = (a - b)2

Hence {(481 + 426)2 - 4 x 481 x 426} = (481 - 426)2 = 552 = 3025

5. (64 - 12)2 + 4 x 64 x 12 = ?A. 5246 B. 4406C. 5126 D. 5776


Answer : Option D

Explanation :

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

Here, the given statement is like (a - b)2 + 4ab where a= 64 and b = 12

(a - b)2 + 4ab = (a2 - 2ab + b2) + 4ab = a2 + 2ab + b2 = (a + b)2

Hence (64 - 12)2 + 4 x 64 x 12 = (64 + 12)2 = 762 = 5776

6. 121 x 54 = ?A. 68225 B. 75625C. 72325 D. 71225


Answer : Option B

Explanation :

121×54=121×(102)4=121×1000016=7.5625×10000=75625

7. If (232 + 1) is completely divisible by a whole number, which of the following numbers is completely divisible by this number?A. (296 + 1) B. (7 x 223 )C. (216 - 1) D. (216 + 1)


Answer : Option A

Explanation :

Let 232 = x. Then (232 + 1) = (x + 1)

Assume that (x + 1) is completely divisible by a whole number, N

(296 + 1) = {(232)3 + 1} = (x3 + 1) = (x + 1)(x2 - x + 1)

if (x + 1) is completely divisible by N, (x + 1)(x2 - x + 1) will also be divisible

by N

Hence (296 + 1) is completely divisible N

8. How many of the following numbers are divisible by 132 ? 264, 396, 462, 792, 968, 2178, 5184, 6336A. 4 B. 3C. 6 D. 8


Answer : Option A

Explanation :

If a number is divisible by two co-prime numbers, then the number is divisible by their

product also.

If a number is divisible by more than two pairwise co-prime numbers,

then the number is divisible by their product also.

Read more

If a number is divisible by another number, then it is also divisible by all the factors

of that number.

Read more

Here 3, 4 and 11 are pairwise co-prime numbers. 132 = 3 × 4 × 11 . Also 3,4 and 11 are

factors of 132. Hence

if a number is divisible by 3, 4 and 11, the number will be divisible by their product 132 also.

If a number is not divisible by 3 or 4 or 11, it is not divisible by 132

You must learn Divisibility Rules to say whether a given number is divisible by another number

without actually performing the division. Please go through divisibility rules before proceeding

further.

264 is divisible by 3, 4 and 11 => 264 is divisible by 132


462 is divisible by 3 and 11, but not divisible by 4 => 462 is not divisible by 132






Hence, only 264, 396 ,792 and 6336 are divisible by 132. So the answer is 4

9. All prime numbers are odd numbersA. True B. False


Answer : Option B

Explanation :

2 is even prime number

10. What is the unit digit in (6324)1797 × (615)316 × (341)476 ?A. 1 B. 2C. 4 D. 0


Answer : Option D

Explanation :

Unit digit in (6324)1797 = Unit digit in (4)1797 = Unit digit in [(42)898 × 4]

= Unit digit in [16898 × 4] = Unit digit in (6 × 4) = 4

Unit digit in (615)316 = Unit digit in (5)316 = 5

Unit digit in (341)476 = Unit digit in (1)476 = 1

Hence, unit digit in (6324)1797 × (625)316 × (341)476 = Unit digit in [4 × 5 × 1] = 0

11. 5216 x 51 = ?A. 266016 B. 212016C. 266436 D. 216314


Answer : Option A

Explanation :

Normal way of multiplication may take time. Here are one alternative.

5216 x 51 = (5216 x 50) + 5216 = (5216 x 100/2) + 5216

= 521600/2 + 5216 = 260800 + 5216 = 266016

12. Which of the following number is divisible by 24 ?A. 31214 B. 61212

C. 512216 D. 3125832Hide Answer

Here is the answer and explanation

Answer : Option D

Explanation :


product also.



Read more


of that number.

Read more

24 = 3 × 8 where 3 and 8 are co-prime numbers. 3 and 8 are also factors of 24. Hence

Hence if a number is divisible by 3, and 8, the number will be divisible by their product 24 also.

If a number is not divisible by 3 or 8, it is not divisible by 24



further.

31214 is not divisible by 3 and 8 => 31214 is not divisible 24

61212 is not divisible by 8 though it is divisible by 3 => 61212 is not divisible 24

512216 is not divisible by 3 though it is divisible by 8 => 512216 is not divisible 24

3125832 is divisible by 3 and 8 => 3125832 is divisible 24

13. 719×719+347×347−719×347719×719×719+347×347×347=?A. 1372 B. 25133C. 11066 D. 56


Answer : Option C

Explanation :

The given statement is in the form a2+b2−aba3+b3 where a = 719 and b = 347

(Reference : Basic Algebraic Formulas)a2+b2−aba3+b3=(a2+b2−ab)(a+b)(a2−ab+b2)=1a+b=1719+347=1719+347=11066

14. If the number 481*673 is completely divisible by 9, what is the the smallest whole number in place of *?A. 3 B. 7C. 5 D. 9


Answer : Option B

Explanation :

Let x be the smallest whole number in place of *

Given that 481*673 to be completely divisible by 9,

=> (4 + 8 + 1 + x + 6 + 7 + 3) is divisible by 9 (Reference : Divisibility by 9)

=> (29 + x) is divisible by 9

x should be the smallest whole number, Hence, (29 + x) = 36

=> x = 36 - 29 = 7

15. If n is a natural number, then (6n2 + 6n) is always divisible by:A. Both 6 and 12 B. 6 onlyC. 12 only D. None of these


Answer : Option A

Explanation :

6n2 + 6n = 6n(n + 1)

Hence 6n2 + 6n is always divisible by 6 and 12 (∵ remember that n(n + 1) is always even)

16. 109 × 109 + 91 × 91 = ?A. 20162 B. 18322C. 13032 D. 18662


Answer : Option A

Explanation :

(a+b)2+(a−b)2=2(a2+b2)

(Reference : Basic Algebraic Formulas)1092+912=(100+9)2+(100−9)2=2(1002+92)=2(10000+81)=20162

17. When (6767 +67) is divided by 68, the remainder isA. 0 B. 22C. 33 D. 66


Answer : Option D

Explanation :

(xn+1) is divisible by (x + 1) only when n is odd

=> (6767 + 1) is divisible by (67 + 1)

=> (6767 + 1) is divisible by 68

=> (6767 + 1) ÷ 68 gives a remainder of 0

=> [(6767 + 1) + 66] ÷ 68 gives a remainder of 66

=> (6767 + 67) ÷ 68 gives a remainder of 66

18. (912+643)2+(912−643)2(912×912+643×643)=?A. 122 B. 2C. 1 D. None of these


Answer : Option B

Explanation :

(a+b)2+(a−b)2=2(a2+b2)

(912+643)2+(912−643)2(912×912+643×643)=(912+643)2+(912−643)2(9122+6432)=2(9122+6432)(9122+6432)=2

19. What is the smallest prime number?A. 0 B. 1C. 2 D. 3


Answer : Option C

Explanation :

smallest prime number is 2.

0 and 1 are neither prime numbers nor composite numbers.

20. (23341379 x 72) = ?A. 1680579288 B. 1223441288C. 2142579288 D. 2142339288

View Answer

21. If the number 5 * 2 is divisible by 6, then * = ?A. 2 B. 7C. 3 D. 5


Answer : Option A

Explanation :

A number is divisible by 6 if it is divisible by both 2 and 3

Read More

Replacing * by x

5 x 2 is divisible by 2 (Reference : Divisibility by 2 rule)

For 5 x 2 to be divisible by 3, 5 + x + 2 shall be divisible by 3 (Reference : Divisibility by 3 rule)

=> 7 + x shall be divisible by 3

=> x can be 2 or 5 or 8

From the given choices, answer = 2

22. (1−1n)+(1−2n)+(1−3n)+...up to n terms=?A. (n−1) B. n2C. 12(n−1) D. 12(n+1)


Answer : Option C

Explanation :

1+1+1+⋯ n terms=∑1=n

1+2+3+⋯+n=∑n=n(n+1)2

(Reference : Click Here)

(1−1n)+(1−2n)+(1−3n)+ ... up to n terms=(1+1+1+ ... up to n terms)−(1n+2n+3n+ ... up to n terms)=n−1n(1+2+3+ ... up to n terms)=n−1n[n(n+1)2]=n−(n+1)2=(2n−n−1)2=n−12

23. What least number should be added to 1056, so that the sum is completely divisible by 23?A. 4 B. 3C. 2 D. 1


Answer : Option C

Explanation :

1056 ÷ 23 = 45 with remainder = 21

Remainder = 21. 21 + 2 = 23. Hence 2 should be added to 1056 so that the

sum will be

divisible by 23

24. 1398 x 1398 = ?A. 1624404 B. 1851404C. 1951404 D. 1954404


Answer : Option D

Explanation :

1398 x 1398

= (1398)2

= (1400 - 2)2

= 14002 - (2 × 1400 × 2) + 22

= 1960000 - 5600 + 4

= 1954404

Note : It will be great if you master speed maths techniques based on Vedic Mathematics,

Trachtenberg System of Mathematics etc so that you can do these calculations even faster

(Reference : Click Here)

25. On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder ?A. 2 B. 3C. 4 D. 5


Answer : Option D

Explanation :

-------------------------------------------------------------------------------------Solution 1-------------------------------------------------------------------------------------Number = 56x + 29 (∵ since the number gives 29 as remainder on dividing by 56)

= (7 × 8 × x) + (3 × 8) + 5

Hence, if the number is divided by 8, we will get 5 as remainder.

-------------------------------------------------------------------------------------Solution 2-------------------------------------------------------------------------------------Number = 56x + 29

Let x = 1. Then the number = 56 × 1 + 29 = 85

85 ÷ 8 = 10, remainder = 5

26. ? + 3699 + 1985 - 2047 = 31111A. 21274 B. 27474C. 21224 D. 27224


Answer : Option B

Explanation :

Let x + 3699 + 1985 - 2047 = 31111

=> x = 31111 - 3699 - 1985 + 2047 = 27474

27. the difference between a positive fraction and its reciprocal is 9/20 find the sum of that fraction and its reciprocal.A. 4120 B. 1720C. 1120 D. 920


Answer : Option A

Explanation :

(Reference : Quadratic Equations and How to Solve Quadratic Equations)

Let the fraction = x

Then x−1x=920⇒x2−1x=920⇒x2−1=9x20⇒20x2−20=9x⇒20x2−9x−20=0x=−b±b2−4ac−−−−−−√2a=9±(−9)2−4×20×(−20)−−−−−−−−−−−−−−−−−−√2×20=9±81+1600−−−−−−−−√40=9±4140=5040 Or −3240Given that the fraction is positive. Hence x=5040=541x=45x+1x=54+45=5×5+4×420=4120

28. How many 3 digit numbers are completely divisible 6 ?A. 146 B. 148C. 150 D. 152


Answer : Option C

Explanation :100/6 = 16, remainder = 4. Hence 2 more should be added to 100 to get the minimum

3 digit number divisible by 6.

=> Minimum 3 digit number divisible by 6 = 100 + 2 = 102999/6 = 166, remainder = 3. Hence 3 should be decreased from 999 to get the maximum

3 digit number divisible by 6.

=> Maximum 3 digit number divisible by 6 = 999 - 3 = 996

Hence, the 3 digit numbers divisible by 6 are 102, 108, 114,... 996

This is Arithmetic Progression with a = 102 ,d = 6, l=996

Number of terms =(l−a)d+1=(996−102)6+1=8946+1=149+1=150

29. How many natural numbers are there between 43 and 200 which are exactly divisible by 6?A. 28 B. 26C. 24 D. 22


Answer : Option B

Explanation :43/6 = 7, remainder = 1. Hence 5 more should be added to 43 to get the minimum

number divisible by 6 between 43 and 200.

=> Minimum number divisible by 6 between 43 and 200 = 43 + 5 = 48200/6 = 33, remainder = 2. Hence 2 should be decreased from 200 to get the maximum

number divisible by 6 between 43 and 200.

=> Maximum number divisible by 6 between 43 and 200 = 200 - 2 = 198

Hence, natural numbers numbers divisible by 6 between 43 and 200 are 48, 54, 60,...198

This is Arithmetic Progression with a = 48, d = 6, l=198

Number of terms =(l−a)d+1=(198−48)6+1=1506+1=25+1=26

30. What is the smallest 6 digit number exactly divisible by 111?A. 100010 B. 100011C. 100012 D. 100013


Answer : Option B

Explanation :

Smallest 6 digit number = 100000100000/111 = 900, remainder = 100. Hence 11 more should be added to 100000

to get the smallest 6 digit number exactly divisible by 111

=> smallest 6 digit number exactly divisible by 111 = 100000 + 11 = 100011

31. If x and y are positive integers such that (3x + 7y) is a multiple of 11, then which of the followings are divisible by 11?A. 9x + 4y B. x + y + 4C. 4x - 9y D. 4x + 6y


Answer : Option C

Explanation :

By hit and trial method, we get x=5 and y=1 such that 3x + 7y = 15 + 7 = 22 is a multiple of 11.

Then

(4x + 6y) = (4 × 5 + 6 × 1) = 26 which is not divisible by 11

(x + y + 4) = (5 + 1 + 4) = 10 which is not divisible by 11

(9x + 4y) = (9 × 5 + 4 × 1) = 49 which is not divisible by 11

(4x - 9y) = (4 × 5 - 9 × 1) = 20 - 9 = 11 which is divisible by 11

32. if (64)2 - (36)2 = 10x, then x = ?A. 200 B. 220C. 210 D. 280

Hide Answer


Answer : Option D

Explanation :

a2−b2=(a−b)(a+b)(64)2 - (36)2 = (64 - 36)(64 + 36) = 28 × 100

Given that (64)2 - (36)2 = 10x

=> 28 × 100 = 10x

=> x = 280

33. 852×852×852−212×212×212852×852+852×212+212×212=?A. 640 B. 620C. 740 D. None of these


Answer : Option A

Explanation :

a3−b3=(a−b)(a2+ab+b2)Given Equation is in the form a3−b3a2+ab+b2 where a = 852 and b = 212

a3−b3a2+ab+b2=(a−b)(a2+ab+b2)(a2+ab+b2)=(a−b)

Hence answer = (a - b) = (852 - 212) = 640

34. 2664÷12÷6=?A. 43 B. 41C. 37 D. 33


Answer : Option C

Explanation :

2664 ÷ 12 = 222

222 ÷ 6 = 37

OR

2664÷12÷6=2664×112×16=37

35. (422+404)2−(4×422×404)=?A. None of these B. 342C. 324 D. 312


Answer : Option C

Explanation :

(a+b)2−4ab=(a2+2ab+b2)−4ab=(a2−2ab+b2)=(a−b)2

Given Equation is in the form (a+b)2−4ab where a = 422 and b = 404

Hence answer = (a+b)2−4ab=(a−b)2=(422−404)2=182=324

36. Which one of the following can't be the square of natural number ?A. 128242 B. 128881C. 130321 D. 131044


Answer : Option A

Explanation :

Square of a natural number cannot end with 2 .

Hence 128242 cannot be the square of natural number

37. (32323 + 7344 + 41330) - (317 x 91) = ?A. 54210 B. 54250C. 52150 D. None of these


Answer : Option C

Explanation :

Its speed and accuracy which decides the winner. You can use various methods including

Vedic mathematics to do complex calculations quickly.

(32323 + 7344 + 41330) - (317 x 91) = 80997 - 28847 = 52150

38. (xn - an) is completely divisible by (x - a), ifA. n is an even natural number B. n is and odd natural numberC. n is any natural number D. n is prime


Answer : Option C

Explanation :

(xn - an) is completely divisible by (x - a) for every natural number n

39. The number 97215*6 is completely divisible by 11. What is the smallest whole number in place of * ?A. 4 B. 2C. 1 D. 3


Answer : Option D

Explanation :

(Reference : Divisibility by 11 rule)

From the above given "Divisibility by 11 rule", it is clear that

if 97215*6 is divisible by 11, then (9 + 2 + 5 + 6) - (7 + 1 + *) is divisible by 11

=> 22 - (8 + *) is divisible by 11

=> (14 - *) is divisible by 11

The smallest whole number which can be substituted in the place of * to satisfy the

above equation is 3 such that 14 - 3 = 11 and 11 is divisible by 11.

Hence the answer is 3

40. (12 + 22 + 32 + ... + 102) = ?A. 395 B. 375C. 55 D. 385


Answer : Option D

Explanation :

(Reference : Power Series : Important formulas)

12+22+32+⋯+n2=∑n2=n(n+1)(2n+1)612+22+32+⋯+102=n(n+1)(2n+1)6=10(10+1)[(2×10)+1]6=10×11×216=10×11×72=385

41. If the product 4864 × 9a2 is divisible by 12, then what is the value of a?A. 1 B. 2C. 5 D. 6


Answer : Option A

Explanation :

(Reference : Divisibility by 12)

A number is divisible by 12 if the number is divisible by both 3 and 4

A number is divisible by 3 if the sum of the digits is divisible by 3

A number is divisible by 4 if the number formed by the last two digits is divisible by 4.

4864 × 9a2 is divisible by 12

=> 4864 × 9a2 is divisible by 3

and 4864 × 9a2 is divisible by 4

4864 is divisible by 4 (Because number formed by the last two digits = 64 which is divisible by 4)

Hence 4864 × 9a2 will also be divisible by 4

4864 is not divisible by 3 (because 4 + 8 + 6 + 4 = 22 which is not divisible by 3)

Hence 9a2 should be divisible by 3 such that 4864 × 9a2 is divisible by 3

=> 9 + a + 2 is divisible by 3

=> 11 + a is divisible by 3

Hence a can be 1 or 4 or 7 such that 11 + a is divisible by 3

So, from the given choices, 1 is the answer

42. -88 × 39 + 312 = ?A. -3120 B. -3200C. 3120 D. 3200


Answer : Option A

Explanation :

Again a question on basic arithmetic. Many algebraic formulas and Vedic Maths can help you

to solve such questions very fast.

-88 × 39 + 312 = -2112 + 312 = -3432 + 312 = -3120

OR

Since 88 can be written as 11 × 8,

-88 × 39 + 312 = -11 × 8 × 39 + 312

= -11 × 312 + 312 (Reference : Multiplication by 11 using Speed Mathematics)

= -3432 + 312 = -3120

OR

-88 × 39 + 312 = -88 × (40-1) + 312

= -88 × 40 + 88 + 312

= -3520 + 88 + 312

= -3120

OR

(using various Speed Mathematics Techniques)

43. 378 × ? = 252A. 23 B. 34C. 12 D. None of these


Answer : Option A

Explanation :

378 × ? = 252

?=252378=126189=1421=23

44. What least number should be subtracted from 13601 such that the remainder is divisible by 87 ?A. 27 B. 28C. 29 D. 30


Answer : Option C

Explanation :

13601 ÷ 87 = 156, remainder = 29

Hence 29 is the least number which can be subtracted from 13601 such that the remainder

is divisible by 87

45. Which one of the given numbers is completely divisible by 45?A. None of these B. 165642C. 202860 D. 112330


Answer : Option C

Explanation :


product also.



Read more


of that number.

Read more

We know that 45 = 9 × 5 where 9 and 5 are co-prime numbers. Also 9 and 5 are


if a number is divisible by 5 and 9, the number will be divisible by their product 45 also.




further.

112330 is divisible by 5 but not divisible by 9 => 112330 is not divisible by 45

202860 is divisible by 5 and 9 => 202860 is divisible by 45

165642 is not divisible by 5 and 9 => 165642 is not divisible by 45

Hence, 202860 is the answer

46. What is the remainder when 17200 is divided by 18 ?A. 3 B. 2C. 1 D. 4


Answer : Option C

Explanation :

(xn−an) is completely divisible by (x+a) when n is even

Read More...(17200 - 1200) is completely divisible by (17 + 1) as 200 is even.

=> (17200 - 1) is completely divisible by 18.

Hence, when 17200 is divided by 18, we will get 1 as remainder.

47. 12 + 22 + 32 + ... + 82 = ?A. 204 B. 200C. 182 D. 214

View Answer

48. 1 + 2 + 3 + ... + 12 = ?A. 66 B. 68C. 76 D. 78


Answer : Option D

Explanation :


1+2+3+⋯+n=∑n=n(n+1)21+2+3+⋯+12=n(n+1)2=12(12+1)2=12×132=6×13=78

49. 13 + 23 + 33 + ... + 63 = ?

A. 451 B. 441C. 421 D. 401


Answer : Option B

Explanation :


13+23+33+⋯+n3=∑n3=n2(n+1)24=[n(n+1)2]2

13+23+33+⋯+63=[n(n+1)2]2=[6(7)2]2=[3×7]2=212=441

50. Which one of the following is a prime number ?A. 307 B. 437C. 247 D. 203


Answer : Option A

Explanation :

307−−−√<18

Prime numbers < 18 are 2, 3, 5, 7, 11, 13, 17

307 is not divisible by 2







Hence 307 is a prime number

437−−−√<21

Prime numbers < 21 are 2, 3, 5, 7, 11, 13, 17, 19








But 437 is divisible by 19 => 437 is not a prime number247−−−√<16

Prime numbers < 16 are 2, 3, 5, 7, 11, 13






But 247 is divisible by 13 => 247 is not a prime number203−−−√<15





But 203 is divisible by 7 => 203 is not a prime number

5

1. The difference of the squares of two consecutive even integers is divisible by which of the following numbers?A. 3 B. 6C. 4 D. 7


Answer : Option C

Explanation :

---------------------------------------------------Solution 1---------------------------------------------------Let the consecutive even integers are n and (n + 2)

difference of the squares = (n + 2)2 - n2 = (n2 + 4n + 4) - n2 = 4n + 4 = 4(n + 1)

which is divisible by 4

---------------------------------------------------Solution 2 (preferred way)---------------------------------------------------Just take any of the two consecutive even integers say 2 and 4

difference of the squares = 42 - 22 = 16 - 4 = 12 which is divisible by 4

52. Which one of the following numbers is completely divisible by 99?A. 115909 B. 115919C. 115939 D. 115929


Answer : Option D

Explanation :


product also.



Read more


of that number.

Read more

We know that 99 = 9 × 11 where 9 and 11 are co-prime numbers. Also 9 and 11 are


if a number is divisible by 9 and 11, the number will be divisible by their product 99 also.




further.

115929 is divisible by both 9 and 11 => 115929 is divisible by 99




Hence, 115929 is the answer

53. 612×612×612+321×321×321612×612−612×321+321×321=?A. 933 B. 1000C. 712 D. 843


Answer : Option A

Explanation :

a3+b3=(a+b)(a2−ab+b2)

Read more...Given Equation is in the form (a3+b3)(a2−ab+b2) where a = 612 and b = 321

(a3+b3)(a2−ab+b2)=(a+b)(a2−ab+b2)(a2−ab+b2)=(a+b)

Hence answer = (a+b)=612+321=933

54. How many terms are there in the G.P. 4, 8, 16, 32, ... , 1024?A. 9 B. 8C. 7 D. 6


Answer : Option A

Explanation :

nth term of a geometric progression (G.P.)

tn=arn−1

where tn = nth term, a= the first term , r = common ratio, n = number of terms

Read more…a=4r=84=2tn=1024tn=arn−1⇒1024=4×2n−1⇒2n−1=10244=256=28⇒2n−1=28⇒n−1=8⇒n=8+1=9

55. 123 × 123 + 288 × 288 + 2 × 123 × 288 = ?A. 168151 B. 178121C. 168921 D. 162481


Answer : Option C

Explanation :

(a+b)2=a2+2ab+b2

Read more...Given Equation is in the form a2+b2+2ab where a = 123 and b = 288

a2+b2+2ab=(a+b)2

Hence answer = (a+b)2=(123+288)2=4112=168921

56. If a number is divided by 6 , 3 is the remainder . What is remainder if the the square of the number is divided by 6?A. 5 B. 4C. 3 D. 2


Answer : Option C

Explanation :

Let the number be x

Given that if the number is divided by 6 , 3 is the remainder

=> x ÷ 6 = k, remainder = 3

Hence x = 6k + 3

Square of the number = x2 = (6k + 3)2

=36k2 + 36k + 9 ∵ (a + b)2 = a2 + 2ab + b2

=36k2 + 36k + 6 + 3

= 6(6k2 + 6k + 1) + 3

Hence, when the square of the number is divided by 6, we get 3 as remainder.

57. In a division sum, the remainder is 0 when a student mistook the divisor by 12 instead of 21 and obtained 35 as quotient. What is the correct quotient ?A. 25 B. 20C. 15 D. 10


Answer : Option B

Explanation :

Let x be the number

x ÷ 12 = 35 , remainder = 0

=> x = 35 × 12

correct quotient = x21=35×1221=5×123=5×4=20

58. 1531 × 132 + 1531 × 68 = ?A. 306000 B. 306100C. 306200 D. 306400

View Answer

59. 100010 ÷ 1028 = ?A. 10 B. 100C. 1000 D. 10000

View Answer

60. 2 + 22 + 222 + 2.22 = ?A. 246 B. 248C. 248.12 D. 248.22


Answer : Option D

61. Which of the following numbers will completely divide (4915 - 1) ?A. 6 B. 7

C. 8 D. 9Hide Answer


Answer : Option C

Explanation :

(xn−an) is completely divisible by (x+a) when n is even

Read More...(4915−1)=[(72)15−1]=(730−1)=(730−130)which is completely divisible by (7+1)=8

62. How many even prime numbers are there less than 50?A. 1 B. 15C. 2 D. 16


Answer : Option A

Explanation :

2 is the only even prime number

63. How many prime numbers are there less than 50 ?A. 13 B. 14C. 15 D. 16


Answer : Option C

Explanation :

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 are the prime numbers less than 50

64. What is the difference between local value and face value of 7 in the numerical 657903?A. 6993 B. 69993C. 7000 D. 7


Answer : Option A

Explanation :

(Local value of 7) - (Face value of 7) = (7000 - 7) = 6993

65. 108 + 109 + 110 + ... + 202 = ?A. 14615 B. 14625C. 14715 D. 14725


Answer : Option D

Explanation :

Number of terms of an arithmetic progression

n=(l−a)d+1

where n = number of terms, a= the first term , l = last term, d= common difference

Read more...Sum of first n terms in an arithmetic progression

Sn=n2[ 2a+(n−1)d ] =n2[ a+l ]where a = the first term, d= common difference, l=tn=nth term = a+(n−1)d

Read more…a=108l=202d=109−108=1n=(l−a)d+1=(202−108)1+1=94+1=95Sn=n2[ a+l ]=952[ 108+202 ]=95×3102=95×155=14725

OR

(Reference : Power Series : Important formulas)1+2+3+⋯+n=∑n=n(n+1)2108+109+110+...+202=[1+2+3+⋯+202]−[1+2+3+⋯+107]=[202×2032]−[107×1082]=[101×203]−[107×54]=20503−5778=14725

66. 23732 × 999 = ?A. 23708268 B. 22608258C. 22608268 D. 23708258


Answer : Option A

Explanation :

23732 × 999

= 23732 × (1000 - 1)

= (23732 × 1000) - (23732 × 1)

= 23732000 - 23732

= 23708268

If you are interested to do fast calculations very fast, please go through speed maths

topics

67. 123427201 - ? = 568794A. 123527207 B. 223521407C. 123527407 D. 122858407


Answer : Option D

Explanation :

Let 123427201 - x = 568794

x = 123427201 - 568794 = 122858407

68. 2210 × ? = 884A. 23 B. 25C. 15 D. 34


Answer : Option B

Explanation :

Let 2210 ×x=884

x=8842210=4421105=3485=25

(Reference : Divisibility Rules)

69. If P and Q are odd numbers, then which of the following is even ?A. P + Q B. PQC. P + Q + 1 D. PQ + 2


Answer : Option A

Explanation :

The sum of two odd numbers is an even number

Read more...

Hence P + Q is an even number

OR

Just take any two odd numbers, say 1 and 3

P + Q = 1 + 3 = 4 = an even number

P + Q + 1 = 1 + 3 + 1 = 5 = an odd number

PQ = 1 × 3 = 3 = an odd number

PQ + 2 = (1 × 3) + 2 = 5 = an odd number

Hence P + Q is the answer

70. Which of the following numbers will completely divide (319 + 320 + 321 + 322) ?

A. 25 B. 16C. 11 D. 12


Answer : Option D

Explanation :

319 + 320 + 321 + 322

= 319(1 + 3 + 32 + 33)

= 319 × 40

= 318 × 3 × 4 × 10

= 318 × 12 × 10 which is divisible by 12

71. What is the largest 5 digit number exactly divisible by 94?A. 99922 B. 99924C. 99926 D. 99928


Answer : Option A

Explanation :

Largest 5 digit number = 99999

99999 ÷ 94 = 1063, remainder = 77

Hence largest 5 digit number exactly divisible by 94 = 99999 - 77 = 99922

72. What is the smallest 5 digit number exactly divisible by 94?A. 10052 B. 10054C. 10056 D. 10058


Answer : Option D

Explanation :

Smallest 5 digit number = 10000

10000 ÷ 94 = 106, remainder = 36

94 - 36 = 58.

ie, 58 should be added to 10000 to make it divisible by 94.

=> smallest 5 digit number exactly divisible by 94 = 10000 + 58 = 10058

73. 1234 - ? = 4234 - 3361A. 351 B. 361C. 371 D. 379


Answer : Option B

Explanation :

Let 1234 - x = 4234 - 3361

x = 1234 - 4234 + 3361 = 361

74. 320 ÷ 2 ÷ 3 = ?A. None of these B. 53.33C. 160 D. 106


Answer : Option B

Explanation :

320÷2÷3=160÷3=53.33

OR

320÷2÷3=320×12×13=1603=53.33

75. 476**0 is divisible by both 3 and 11. What are the non-zero digits in the hundred's and ten's places respectively?A. 8 and 5 B. 6 and 5C. 8 and 2 D. 6 and 2


Answer : Option A

Explanation :

References

1. Divisibility by 3


-----------------------------------------------------------------------------

Solution 1(Recommended) - hit and trial method

-----------------------------------------------------------------------------

Just substitute the values in the missing places and apply divisibility rules.

if a=6 and b=2, number = 476620

4 + 7 + 6 + 6 + 2 + 0 = 25 which is not divisible by 3

Hence 476620 is not divisible by 3


4 + 7 + 6 + 8 + 2 + 0 = 27 which is divisible by 3

Hence 476820 is divisible by 3

4 + 6 + 2 = 12

7 + 8 + 0 = 15

15 - 12 = 3



4 + 7 + 6 + 6 + 5 + 0 = 28 which is not divisible by 3



4 + 7 + 6 + 8 + 5 + 0 = 30 which is divisible by 3


4 + 6 + 5 = 15

7 + 8 + 0 = 15

15 - 15 = 0


So a=8 and b=5 is the answer

-----------------------------------------------------------------------------

Solution 2

-----------------------------------------------------------------------------

Let the number be 476ab0

476ab0 is divisible by 3

=> 4 + 7 + 6 + a + b + 0 is divisible by 3

=> 17 + a + b is divisible by 3 ------------------------(Equation 1)

476ab0 is divisible by 11

[(4 + 6 + b) -(7 + a + 0)] is 0 or divisible by 11

=> [3 + (b - a)] is 0 or divisible by 11 ------------------------(Equation 2)

Substitute the values of a and b with the values given in the choices and select the

values which satisfies both Equation 1 and Equation 2.

if a=6 and b=2,

17 + a + b = 17 + 6 + 2 = 25 which is not divisible by 3 --- Does not meet equation 1

Hence this is not the answer

if a=8 and b=2,

17 + a + b = 17 + 8 + 2 = 27 which is divisible by 3 --- meet equation 1

[3 + (b - a)] = [3 + (2 - 8)] = -3 which is neither 0 nor divisible by 11---Does not meet equation 2


if a=6 and b=5,

17 + a + b = 17 + 6 + 5 = 28 which is not divisible by 3 --- Does not meet equation 1


if a=8 and b=5,

17 + a + b = 17 + 8 + 5 = 30 which is divisible by 3 --- meet equation 1

[3 + (b - a)] = [3 + (5 - 8)] = 0 ---meet equation 2

Since these values satisfies both equation 1 and equation 2, this is the answer

76. On dividing 2272 as well as 875 by 3-digit number N, we get the same remainder. What is the sum of the digits of N?A. 11 B. 10C. 9 D. 8


Answer : Option B

Explanation :

Let 2272 ÷ N = a, remainder = r

=> 2272 = Na + r ----------------------------(Equation 1)

Let 875 ÷ N = b, remainder = r

=> 875 = Nb + r ----------------------------(Equation 1)

(Equation 1) - (Equation 2)

=> 2272 - 875 = [Na + r] - [Nb + r] = NA - Nb = N(a - b)

=> 1397 = N(a - b) ----------------------------(Equation 3)

It means 1397 is divisible by N

But 1397 = 11 × 127

[Reference1 :how to find factors of a number?]

[Reference2 :Prime Factorization?]

You can see that 127 is the only 3 digit number which perfectly divides 1397

=> N = 127

sum of the digits of N = 1 + 2 + 7 = 10

77. What is the sum of first ten prime numbers?A. 55 B. 101C. 130 D. 129


Answer : Option D

Explanation :

(Reference : Prime Numbers)

Required Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129

78. Which f the following numbers is exactly divisible by 11?A. 499774 B. 47554C. 466654 D. 4646652


Answer : Option A

Explanation :

Reference : Divisibility by 11 Rule

Take 47554

4 + 5 + 4 = 13

7 + 5 = 12

13 - 12 = 1



Take 466654

4 + 6 + 4 = 14

6 + 5 = 11

14 - 11 = 3



Take 4646652

4 + 4 + 6 + 2 = 16

6 + 6 + 5 = 17

17 - 16 = 1



Take 499774

4 + 9 + 7 = 20

9 + 7 + 4 = 20

20 - 20 = 0

We got the difference as 0.


79. What is the sum all even natural numbers between 1 and 101?A. 5050 B. 2550C. 5040 D. 2540


Answer : Option B

Explanation :

Reference1 : Natural Numbers

Reference2 : Arithmetic Progression (AP) and Related Formulas

Required sum = 2 + 4 + 6+ . . . + 100

This is an arithmetic progression with

a = 2

d = (4 - 2) = 2

n=(l−a)d+1=(100−2)2+1=982+1=49+1=50

2+4+6+⋯+100=n2(a+l)=502(2+100)=502(102)=50×51=2550

80. A boy multiplies 987 by a certain number and obtained 559981 as his answer. If in the answer both 9 are wrong , but the other digits are correct, then what will be the correct answer?A. 556581 B. 555681C. 555181 D. 553681


Answer : Option B

Explanation :

-----------------------------------------------------------------------------Solution 1

-----------------------------------------------------------------------------The answer is divisible by 987.

So we can use hit and trial method to find out the number divisible by 987

from the given choices

553681 ÷ 987 gives a remainder not equal to 0



But 555681 ÷ 987 gives 0 as remainder. Hence this is the answer

-----------------------------------------------------------------------------Solution 2 (Recommended)

-----------------------------------------------------------------------------The answer is divisible by 987.


product also.



Read more


of that number.

Read more

987 = 3 × 7 × 47 [Prime Factorization]

Here 3, 7 and 47 are pairwise co-prime numbers.

Also 3, 7 and 47 are factors of 987. Hence

if a number is divisible by 3, 7 and 47, the number will be divisible by their product 987 also.

If a number is not divisible by 3 or 7 or 47, it is not divisible by 987



further.

556581 is divisible by 3










Hence 555681 is divisible by 987 also

81. Which one of the following cannot be the square of natural number ?A. 15186125824 B. 49873162329C. 14936506225 D. 60625273287


Answer : Option D

Explanation :

Square of a natural number cannot end with 7. Hence 60625273287 is the answer

82. 7128 + 1252 = 1202 + ?A. 6028 B. 1248

C. 2348 D. 7178Hide Answer


Answer : Option D

Explanation :

? = 7128 + 1252 - 1202 = 7128 + 50 = 7178

83. 3 + 32 + 33 + ... + 38 = ?A. 9820 B. 9240C. 9840 D. 9220


Answer : Option C

Explanation :

Sum of first n terms in a geometric progression (G.P.)

Sn=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪a(rn−1)r−1 a(1−rn)1−r (if r > 1)(if r < 1)

where a= the first term , r = common ratio, n = number of terms

Read more...This is a Geometric Progression (GP) where

a = 3r=323=3

n = 8Sn=a(rn−1)r−1=3(38−1)3−1=3(38−1)2=3(6561−1)2=3×65602=3×3280=9840

84. 73411 × 9999 = ?A. 724836589 B. 724036589

C. 734036589 D. 734036129Hide Answer


Answer : Option C

Explanation :

73411 × 9999 = 73411(10000 - 1) = 734110000 - 73411 = 734036589

85. 32 + 33 + 34 + ... + 42 = ?A. 397 B. 407C. 417 D. 427


Answer : Option B

Explanation :


n=(l−a)d+1




Read more…a=32l=42d=33−32=1n=(l−a)d+1=(42−32)1+1=10+1=11Sn=n2[ a+l ]=112[ 32+42 ]=11×742=11×37=407

OR

(Reference : Power Series : Important formulas)1+2+3+⋯+n=∑n=n(n+1)232+33+34+...+42=[1+2+3+⋯+42]−[1+2+3+⋯+31]=[42×432]−[31×322]=[21×43]−[31×16]=903−496=407

86. What is the digit in the unit place of the number represented by (795 - 358)A. 4 B. 3C. 2 D. 1


Answer : Option A

Explanation :

First let's find out the unit digit of 795

795 = [(74)23 × 73]

Hence, Unit Digit of 795

= Unit Digit of (74)23 × Unit Digit of 73 ----(Equation 1)

Unit Digit of [(74)23]

= Unit Digit of [(7×7×7×7)23]

= Unit Digit of [(9 × 9)23] (∵ 7×7=49 and 9 is the unit digit of 49)

= Unit Digit of [123] (∵ 9×9=81 and 1 is the unit digit of 81)

= 1 ----(Equation 2)

Unit Digit of 73

= Unit Digit of [7 × 7 × 7]

= Unit Digit of [9 × 7](∵ 7 × 7 = 49 and 9 is the unit digit of 49)

= 3(∵ 9 × 7 = 63 and 3 is the unit digit of 63) ----(Equation 3)

Hence from Equation 1,Equation 2 and Equation 3,

unit digit of 795 = 1 × 3 = 3 ----(Equation A)

Similarly we can find out unit digit of 358

358 = [(34)14 × 32]




= Unit Digit of [(3×3×3×3)14]

= Unit Digit of [(9 × 9)14] (∵ 3×3=9)



Unit Digit of 32 = 9----(Equation 6)


Unit Digit of 358 = 1 × 9 = 9 ----(Equation B)

We have already found out that

Unit Digit of 795 = 3 (From equation A)

and Unit Digit of 358 = 9 (From equation B)

Hence, Unit Digit of (795 - 358)

= Unit Digit of 795 - Unit Digit of 358

= unit digit of [larger number of last digit 3 - smaller number of last digit 9 ](∵ 795 > 358)

= 4 (∵ 4 is the unit digit when a smaller number of

last digit 9 is subtracted from a larger number of last digit 3. example : 113 - 19 = 94)

87. What is the unit digit in the number represented by [365 × 659 × 771]A. 1 B. 2C. 3 D. 4


Answer : Option D

Explanation :

First let's find out the unit digit of 365

365 = [(34)16 × 3]


= Unit Digit of (34)16 × 3

= Unit Digit of [(3×3×3×3)16] × 3

= Unit Digit of [(9 × 9)16] × 3 (∵ 3×3=9)

= Unit Digit of [116] × 3 (∵ 9×9=81 and 1 is the unit digit of 81)

= 1 × 3

= 3 ----(Equation A)

Unit digit of 659 = 6 (∵ 6×6=36 and unit digit remains as 6 always) ----(Equation B)

771 = [(74)17 × 73]




= Unit Digit of [(7×7×7×7)17]

= Unit Digit of [(9 × 9)17] (∵ 7×7=49 and 9 is the unit digit of 49)



Unit Digit of 73

= Unit Digit of [7 × 7 × 7]

= Unit Digit of [9 × 7](∵ 7 × 7 = 49 and 9 is the unit digit of 49)

= 3(∵ 9 × 7 = 63 and 3 is the unit digit of 63) ----(Equation 3)


unit digit of 771 = 1 × 3 = 3 ----(Equation C)

We have already found out that

Unit Digit of 365 = 3 (From equation A)

Unit Digit of 659 = 6 (From equation B)

Unit Digit of 771 = 3 (From equation C)

Hence, unit digit in the number represented by [365 × 659 × 771]

= unit digit of [3 × 6 × 3]

= unit digit of [8 × 3](∵ 3 × 6 = 18 and 8 is the unit digit of 18)

= 4 (∵ 8 × 3 = 24 and 4 is the unit digit of 24)

88. 112 × 112 + 88 × 88 = ?A. 26218 B. 20328C. 20288 D. 24288


Answer : Option C

Explanation :

(a+b)2=a2+2ab+b2

(a−b)2=a2−2ab+b2

(a+b)2+(a−b)2=2(a2+b2)

Read more...112×112+88×88=1122+882=(100+12)2+(100−12)2=2(1002+122)=2(10000+144)=2×10144=20288

89. 9312 x 9999 = ?A. 93110688 B. 93010688C. 93110678 D. 83110688


Answer : Option A

Explanation :

9312 x 9999 = 9312(10000 - 1) = 93120000 - 9312 = 93110688

90. 112 × 112 - 88 × 88 = ?A. 4600 B. 4700C. 4800 D. 4900


Answer : Option C

Explanation :

a2−b2=(a−b)(a+b)

Read more...112×112−88×88=1122−882=(112−88)(112+88)=24×200=4800

91. 1234 + 123 + 12 - ? = 1221A. 148 B. 158C. 168 D. 178


Answer : Option A

Explanation :

? = 1234 + 123 + 12 - 1221 = 148

92. A three-digit number 4a3 is added to another three-digit number 984 to give a four digit number 13b7, which is divisible by 11. What is the value of (a + b)?A. 9 B. 10C. 11 D. 12


Answer : Option B

Explanation :

(Reference : Divisibility by 11 rule)

4 a 3

9 8 4

----------

13 b 7

=> a + 8 = b -----------(Equation 1)

13b7 is divisible by 11

=> (1 + b) - (3 + 7) is 0 or divisible by 11

=> (b - 9) is 0 or divisible by 11 -----------(Equation 2)

Assume that (b - 9) = 0

=> b = 9

Substituting the value of b in Equation 1,

a + 8 = b

a + 8 = 9

=> a = 9 - 8 = 1

If a = 1 and b= 9,

(a + b) = 1 + 9 = 10

10 is there in the given choices. Hence this is the answer.

93. 122 × 122 + 322 × 322 - 2 × 122 × 322 = ?A. 44000 B. 42000C. 38000 D. 40000


Answer : Option D

Explanation :

(a−b)2=a2−2ab+b2

Read more...122×122+322×322−2×122×322=1222−2×122×322+3222=(122−322)2=(−200)2=40000

94. [22 + 42 + 62 + ... + 142) = ?A. 559 B. 363C. 364 D. 560


Answer : Option D

Explanation :

12+22+32+⋯+n2=∑n2=n(n+1)(2n+1)6

Read More ...

22+42+62+⋯+142=(2×1)2+(2×2)2+(2×3)2+⋯+(2×7)2=(22×12)+(22×22)+(22×32)+⋯+(22×72)=22(12+22+32+⋯+72)=22[n(n+1)(2n+1)6]=4[7(7+1)[(2×7)+1]6]=4×7×8×156=4×7×8×52=2×7×8×5=56×10=560

95. The sum of the two numbers is 11 and their product is 24. What is the sum of the reciprocals of these numbers ?A. 712 B. 1112C. 1124 D. 78


Answer : Option C

Explanation :

Let the numbers be x and y. Thenx+y=11xy=24Hence,x+yxy=1124⇒1y+1x=1124

96. What is the difference between the place values of two sevens in the numeral 54709479 ?A. 699930 B. 699990C. 99990 D. None of these


Answer : Option A

Explanation :

Required Difference = 700000 - 70 = 699930

97. Which of the following numbers completely divides (461 + 462 + 463 + 464) ?A. 11 B. 10C. 9 D. 8

Hide Answer


Answer : Option B

Explanation :

461 + 462 + 463 + 464

= 461[1 + 41 + 42 + 43]

= 461[1 + 4 + 16 + 64]

= 461 × 85

= 461 × 5 × 17

= 460 × 4 × 5 × 17

= 460 × 20 × 17

= 460 × 10 × 2 × 17

Hence this is completely divisible by 10

98. A number when divided by 75 leaves 34 as remainder. What will be the remainder if the same number is divided by 65?A. 3 B. 1C. 6 D. 9


Answer : Option D

Explanation :

Let the number be x

Let x ÷ 75 = p and remainder = 34

=> x = 75p + 34

= (25p × 3) + 25 + 9

= 25(3p + 1) + 9

Hence, if the number is divided by 25, we will get 9 as remainder

99. The number 7490xy is divisible by 90. Find out (x + y).A. 4 B. 5C. 6 D. 7


Answer : Option D

Explanation :


of that number.

Read more

A number is divisible by 10 if the last digit is 0. Read More

A number is divisible by 9 if the sum of its digits is divisible by 9. Read More

90 = 10 × 9

7490xy is divisible by 90. Since 10 and 9 are factors of 90,

7490xy is divisible by 10 and 7490xy is divisible by 9

7490xy is divisible by 10. We know that A number is divisible by 10 if the last digit is 0.

Hence y = 0

Thus we have the number 7490x0 which is divisible by 9

=> sum of its digits of 7490x0 is divisible by 9

=> 7 + 4 + 9 + 0 + x + 0 is divisble by 9

=> 20 + x is divisible by 9 (Where x is a digit)

=> x = 7

Hence, (x + y) = (7 + 0) = 7

100. What is the smallest 3 digit prime number?A. 107 B. 100C. 102 D. 101


Answer : Option D

Explanation :

1. Prime Numbers

2. Divisibility Rules

102 is divisible by 2 => 102 is not a prime number

100 is divisible by 2 => 100 is not a prime number

101−−−√<11

101 is not divisible by the prime numbers 2, 3, 5, 7.

Hence 101 is a prime number

Since 100 is not a prime number, 101 is the smallest 3 digit prime number

107−−−√<11

107 is also not divisible by the prime numbers 2, 3, 5, 7.

Hence 107 is also a prime number.

But the smallest 3 digit prime number is 101

101. Which is the natural number nearest to 11720 and completely divisible by 58?A. 11716 B. 11712C. 11718 D. 11714


Answer : Option A

Explanation :

11720 ÷ 58 = 202, remainder = 4

Hence the natural number nearest to 11720 and completely divisible by 58

= 11720 - 4 = 11716

102. 563124555 - ? = 232323A. 562892232 B. 562892222C. 562892212 D. 562892202


Answer : Option A

Explanation :

? = 563124555 - 232323 = 562892232

103. What is the sum of first 200 natural numbers?A. 20120 B. 19901C. 19900 D. 20100


Answer : Option D

Explanation :

1+2+3+⋯+n=∑n=n(n+1)2

Read More...Required Sum = 1+2+3+...+200=n(n+1)2=200(200+1)2=200×2012=100×201=20100

OR


n=(l−a)d+1




Read more…Required Sum, Sn=1+2+3+...+200a=1l=200d=2−1=1n=200Sn=n2[ a+l ]=2002[ 1+200 ]=100×201=20100

104. The number 1242*2 is completely divisible by 3. What is the smallest number in place of * ?A. 3 B. 2C. 1 D. 0


Answer : Option C

Explanation :

A number is divisible by 3 if the sum of the digits is divisible by 3

Read more

1242*2 is divisible by 3

=> 1 + 2 + 4 + 2 + * + 2 is divisible by 3

=> 11 + * is divisible by 3

We need to find out the smallest value of * which satisfies the above equation.

Hence, the smallest value of * is 1 such that 11 + 1 = 12 which is divisible by 3

105. Which of the given numbers is divisible by 3, 7, 9 and 11?A. None of these B. 1890C. 4230 D. 6237


Answer : Option D

Explanation :





1. Testing 4230

4 + 2 + 3 + 0 = 9. 9 is divisible by 3. Hence 4230 is also divisible by 3


423 - (2 × 0) = 423

42 - (2 × 3) = 36

36 is not divisible by 7. Hence 4230 is not divisible by 7

Hence 4230 does not meet all divisibility conditions

2. Testing 1890



189 - (2 × 0) = 189

18 - (2 × 9) = 0


1 + 9 = 10

8 + 0 = 8

10 - 8 = 2

2 is not divisible by 11. Hence 1890 is not divisible by 11

Hence 1890 does not meet all divisibility conditions

2. Testing 6237



623 - (2 × 7) = 609

60 - (2 × 9) = 42

42 is divisible by 7. Hence 6237 is also divisible by 7.

6 + 3 = 9

2 + 7 = 9

9 - 9 = 0


We got that 6237 is divisible by 3, 9, 7 and 11. Hence this is the answer.

106. 17 + 16 × 1.6 + 14 × 1.3 = ?A. 60.8 B. 68.8C. 60.6 D. 59.6


Answer : Option A

Explanation :

17 + 16 × 1.6 + 14 × 1.3

= 17 + 25.6 + 18.2

= 60.8

107. 143×7298= ?A. 1642 B. 1802C. 2022 D. 1372


Answer : Option D

Explanation :

143×7298=143×7214×7=142×7=196×7=1372

108. What smallest number should be added to 8444 such that the sum is completely divisible by 7 ?A. 6 B. 5C. 4 D. 3


Answer : Option B

Explanation :

8444 ÷ 7 = 1206, remainder = 2

7 - 2 = 5

Hence, 5 should be added to 8444 such that the sum is completely divisible by 7.

109. 5332 × 992 = ?A. 5289344 B. 5289244C. 5289214 D. 5289324


Answer : Option A

Explanation :

5332 × 992 = 5332(1000 - 8) = 5332000 - 42656 = 5289344

You can find many shortcuts to do calculations with in seconds. It is strongly

recommend to go through Speed Mathematics which will save a lot of time when doing calculations.

110. 40% of 2/3 of a number is 32. What is the number?A. 160 B. 240C. 80 D. 120


Answer : Option D

Explanation :

Let x be the number. Thenx×23×40100=32⇒x=32×10040×32=32×104×32=8×10×32=4×10×3=120

111. If a whole number n is divided by 4, we will get 3 as remainder. What will be the remainder if 2n is divided by 4 ?A. 4 B. 3C. 2 D. 1


Answer : Option C

Explanation :

Let n ÷ 4 = p , remainder = 3

=> n = 4p + 3

2n = 2(4p + 3) = 8p + 6 = 8p + 4 + 2 = 4(2p + 1) + 2

Hence, if 2n is divided by 4, we will get 2 as remainder.

112. 241 × 999 = ?A. 240769 B. 230759C. 230769 D. 240759


Answer : Option D

Explanation :

241 × 999 = 241(1000 - 1) = 241000 - 241 = 240759

You can find many shortcuts to do calculations with in seconds. It is strongly

recommend to go through Speed Mathematics which will save a lot of time when doing calculations.

113. The number 367505*8 is completely divisible by 8. What is the smallest whole number in place of *?A. 1 B. 2C. 3 D. 4


Answer : Option B

Explanation :

A number is divisible by 8 if the number formed by the last three digits is divisible by 8.

Read more


=> 5*8 is divisible by 8

We need to find out the smallest value of * which satisfies the condition 5*8 is divisible by 8


528 is divisible by 8. Hence the smallest value of * is 2 which satisfies the condition


114. 425 × ? = 170A. 12 B. 25C. 35 D. 23


Answer : Option B

Explanation :

425 × ? = 170

⇒? = 170425=3485=25

115. The unit digit in 7105 isA. 8 B. 7C. 6 D. 5


Answer : Option B

Explanation :

7105 = (74)26 × 7

Hence, Unit digit of 7105

= Unit digit of [(74)26 × 7]

= Unit digit of [(7 × 7 × 7 × 7)26 × 7]

= Unit digit of [(9 × 9)26 × 7](∵ 7 × 7 = 49 and 9 is the unit digit of 49)

= Unit digit of [126 × 7](∵ 9 × 9 = 81 and 1 is the unit digit of 81)

= Unit digit of [1 × 7](∵ Unit digit of 126 = 1)

= 7

116. Which of the following numbers is a prime number ?A. None of these B. 377C. 469 D. 176


Answer : Option A

Explanation :

Reference : Divisibility Rules

469−−−√<22





But 469 is divisible by 7

Hence 469 is not a prime number176−−−√<14



Hence 176 is not a prime number377−−−√<20







But 377 is divisible by 13

Hence 377 is not a prime number

Hence answer is "None of these"

117. In a division sum, the divisor is 10 times the quotient and 5 times the remainder. If the remainder is 46, find out the dividend.A. 4426 B. 3426C. 4336 D. 5336

View Answer

118. The difference of two numbers is 1365. On dividing the larger number by the smaller, 6 is obtained as quotient and 15 as remainder. What is the smaller number ?A. 310 B. 330C. 250 D. 270


Answer : Option D

Explanation :

Let the smaller number be x

and the bigger number be (x + 1365)

(x + 1365) ÷ x = 6, remainder = 15

=> (x + 1365) = 6x + 15

=> 5x = 1350

=> x = 1350/5 = 270

Smaller number = x = 270

119. Difference between the squares of two consecutive odd integers is always divisible byA. 8 B. 7C. 6 D. 3


Answer : Option A

Explanation :

-----------------------------------------------------------------Solution 1------------------------------------------------------------------Let two odd numbers be (2n - 1) and (2n + 1)

Difference between the squares

= (2n + 1)2 - (2n - 1)2

= [4n2 + 4n + 1] - [4n2 - 4n + 1]

= 8n which is always divisible by 8

-----------------------------------------------------------------Solution 2------------------------------------------------------------------Take any two odd numbers, say 1 and 3

Difference between the squares = 32 - 12 = 9 - 1 = 8

From the given choices, now you can easily figure out the answer as 8

120. A number was divided successively in order by 4, 5 and 6. The remainders were 2, 3 and 4 respectively. What is the number?A. 224 B. 324C. 304 D. 214


Answer : Option D

Explanation :

Let P be the number

Given that

P ÷ 4 = Q, remainder = 2

Q ÷ 5 = R, remainder = 3

R ÷ 6 = 1, remainder = 4

Hence R = 1 × 6 + 4 = 10

Q = 5R + 3 = (5 × 10) + 3 = 53

P = 4Q + 2 = (4 × 53) + 2 = 212 + 2 = 214

121. What is the sum of first 20 natural numbers?A. 220 B. 205C. 190 D. 210


Answer : Option D

Explanation :

1+2+3+⋯+n=∑n=n(n+1)2

Read More...Required Sum = 1+2+3+...+20=n(n+1)2=20(20+1)2=20×212=10×21=210

OR


n=(l−a)d+1


Read more...

Sum of first n terms in an arithmetic progression


Read more…Required Sum, Sn=1+2+3+...+20a=1l=20d=2−1=1n=20Sn=n2[ a+l ]=202[ 1+20 ]=10×21=210

122. ? - 23442 - 12411 = 2469A. 38322 B. 37212C. 37532 D. 38122

Show Answer

123. In dividing a number by 585, a student employed the method of short division. He divided the number successively by 5, 9 and 13 and got the remainders 4, 8, 12 respectively. What would have been the remainder if he had divided the number by 585?A. 144 B. 292C. 24 D. 584


Answer : Option D

Explanation :

Let P be the number

Given that


Q ÷ 9 = R, remainder = 8

R ÷ 13 = 1, remainder = 12

Hence R = 1 × 13 + 12 = 25

Q = 9R + 8 = (9 × 25) + 8 = 225 + 8 = 233

P = 5Q + 4 = (5 × 233) + 4 = 1165 + 4 = 1169

P ÷ 585 = 1169 ÷ 585 = 1, remainder = 584

124. A number when divided successively by 4 and 5 leaves remainders 1 and 4 respectively. What will be the respective remainders if it is successively divided by 5 and 4?A. 1,2 B. 2, 3C. 3, 2 D. 2,1


Answer : Option B

Explanation :

Let P be the number

Given that


Q ÷ 5 = 1, remainder = 4

Hence Q = 1 × 5 + 4 = 9

P = 4Q + 1 = (4 × 9) + 1 = 36 + 1 = 37

Now divide 37 successively by 5 and 4 respectively

37 ÷ 5 = 7, remainder = 2

7 ÷ 4 = 1, remainder = 3

125. What is the sum of even numbers between 9 and 53?A. 682 B. 672C. 662 D. 702


Answer : Option A

Explanation :


n=(l−a)d+1




Read more…Required Sum =10+12+14+⋯+52This is an Arithmetic Progression with a=10l=52d=12−10=2n=(l−a)d+1=(52−10)2+1=21+1=22Sn=n2[ a+l ]=222[ 10+52 ]=11×62=682

126. The number 13*48 is exactly divisible by 72. Find out the minimum value of * .A. 1 B. 2C. 3 D. 4


Answer : Option B

Explanation :


of that number.

Read more

A number is divisible by 8 if the number formed by the

last three digits is divisible by 8. Read More


72 = 8 × 9

Hence 8 and 9 are factors of 72

Hence, if the a number is divisible by 72, it must be divisible by 8 and 9 also

13*48 is divisible by 72. Since 9 and 8 are factors of 72,

13*48 is divisible by 9 and 13*48 is divisible by 8

13*48 is divisible by 9.

=> 1 + 3 + * + 4 + 8 is divisible by 9

=> 16 + * is divisible by 9 -----------------------(Equation 1)


=> *48 is divisible by 8

=> * can be 2 or 4 or 6 or 8 -----------------------(Equation 2)

We need to find out the minimum value of * which satisfies both equation 1 and 2

Take the value * = 2 from Equation 2

16 + * = 16 + 2 = 18 is divisible by 9.

Hence * = 2 is the minimum value which satisfies both equation 1 and 2

and this is the answer

OR

Use hit and trial method


product also.

Read more

Here 9 and 8 are pairwise co-prime numbers. 72 = 9 × 8.

Hence, if a number is divisible by 9 and 8, the number will be divisible by their

product 72 also.

1 is the minimum value given as the choices

Substituting 1 in the place of *, we get 13148

13148 is not divisible by 9 => 13148 is not divisible by 72

Substituting 2 in the place of *, we get 13248




Hence the minimum value of * is 2

127. Which of the following number is divisible by 3 but not by 9 ?A. 5271 B. 4122C. 3141 D. 3222


Answer : Option A

Explanation :

A number is divisible by 3 if the sum of the digits is divisible by 3 Read More


Take 3222

3 + 2 + 2 + 2 = 9 which is divisible by 3 and 9.

Hence 3222 is divisible by 3 and 9

Take 3141



Take 4122



Take 5271

5 + 2 + 7 + 1 = 15 which is divisible by 3 , but not divisible by 9.

Hence 5271 is divisible by 3 , but not divisible by 9

128. When a number is divided by 13, the remainder is 6. When the same number is divided by 7, then remainder is 1. What is the number ?A. 243 B. 253C. 312 D. None of these


Answer : Option B

Explanation :

Take 243

243 ÷ 7 = 34, Remainder = 5


Take 312

312 ÷ 7 = 44, Remainder = 4


Take 253

253 ÷ 7 = 36, Remainder = 1 .

253 ÷ 13 = 19, Remainder = 6.

This satisfies both the conditions given in the question. Hence this is the answer.

129. What is the sum of all two digit numbers divisible by 6?A. 805 B. 820C. 790 D. 810


Answer : Option D

Explanation :


n=(l−a)d+1




Read more…Required Sum =12+18+24+⋯+96This is an Arithmetic Progression with a=12l=96d=6n=(l−a)d+1=(96−12)6+1=14+1=15Sn=n2[ a+l ]=152[ 12+96 ]=15×1082=15×54=810

130. 2002 × 2002 = ?A. 4008004 B. 4006004C. 4002004 D. 4004004


Answer : Option A

Explanation :

2002 × 2002 = 4008004

OR

(a+b)2=a2+2ab+b2

Read more...2002 × 2002 = 20022

= (2000 + 2)2

= 2000 2 + (2 × 2000 × 2) + 22

= 4000000 + 8000 + 4

= 4008004

OR

Using speed mathematics techniques given at Speed Mathematics

which will save a lot of time when doing calculations.

131. Which natural number is completely divisible by 123 and nearest to 410081A. 410082 B. 409959C. 410078 D. 410071

Hide Answer


Answer : Option A

Explanation :

410081 ÷ 123 = 3333, remainder = 122

Hence required number = 410081 + (123 - 122) = 410082

132. What is the difference between the place value and the face value of 6 in the numeral 296827?A. None of these B. 5999C. 994 D. 5994


Answer : Option D

Explanation :

Place value of 6 = 6000

Face value of 6 = 6

Difference = 6000 - 6 = 5994

133. The sum of a series, 27 + 36 + 45 + ... + 162 is 1512. What is the number of terms in the series?A. 14 B. 15C. 16 D. 17


Answer : Option C

Explanation :

Sum of first n terms in an arithmetic progression

Sn=n2[ 2a+(n−1)d ] =n2[ a+l ]where a = the first term, d= common

difference, l=tn=nth term = a+(n−1)d

Read more...a=27d=36−27=9Sn=27+36+45+...+162=1512⇒n2[ 2a+(n−1)d ]=1512⇒n2[ (2×27)+(n−1)9 ]=1512⇒n2[ 54+9n−9 ]=1512⇒n(45+9n)=3024⇒9n(5+n)=3024⇒n(5+n)=336 -------(Equation 1)

From here, you can either solve the quadratic equation using the techniques given here

or just solve use hit and trial method ; both are given below.

--------------------------------------------------------------------------------------

Solution 1 - Solving the quadratic equation n(5 + n) = 336 to find out the value of n

--------------------------------------------------------------------------------------

n(5+n)=336n2+5n−336=0n=−b±b2−4ac−−−−−−√2a=−5±52−[4×1×(−336)]−−−−−−−−−−−−−−−−√(2×1)=−5±(25+1344)−−−−−−−−−√2=−5±1369−−−−√2=−5±372=−5+372 (∵ since n is the number of terms, only positive value is applicable here)=322=16=Hence, number of terms = 16

--------------------------------------------------------------------------------------

Solution 2 - Solving the quadratic equation using hit and trial method

--------------------------------------------------------------------------------------

Take Equation 1 : n(5 + n) = 336

Take the values given in the choices and see which value satisfies the above equation

If n = 14, n(5 + n) = 14 × 19 = 266 ? 336

If n = 15, n(5 + n) = 15 × 20 = 300 ? 336

If n = 16, n(5 + n) = 16 × 21 = 336.

Hence n = 16 satisfies equation 1.

Hence, number of terms = 16

134. 5+14−3+23+1−35= ?A. 18960 B. 19960C. 12960 D. 16960


Answer : Option B

Explanation :

5+14−3+23+1−35=(5−3+1)+(14+23−35)=3+(1×15+2×20−3×12)60=3+1960=180+1960=19960

135. 996ab is divisible by 80. What is (a + b) ?A. 3 B. 5C. 6 D. 8


Answer : Option D

Explanation :


of that number.

Read more

A number is divisible by 8 if the number formed by the last three digits is divisible by 8. Read More


80 = 10 × 8

996ab is divisible by 80. Since 10 and 8 are factors of 80,

996ab is divisible by 10 and 996ab is divisible by 8

996ab is divisible by 10. We know that A number is divisible by 10 if the last digit is 0.

Hence b = 0

Thus we have the number 996a0 which is divisible by 8

=> 6a0 is divisible by 8

=> a = 0 or 4 or 8

Hence, (a + b)

= (0 + 0) or (4 + 0) or (8 + 0)

= 0 or 4 or 8

136. 123 - 333 + 321 - 111 = ?A. 320 B. 100C. 120 D. 0


Answer : Option D

Explanation :

123 - 333 + 321 - 111 = 444 - 444 = 0

137. On multiplying a number by 3, the product is a number each of whose digits is 7. What is the smallest such number?A. 259129 B. 259219C. 259279 D. 259259


Answer : Option D

Explanation :

Let's use hit and trial method here

Smallest number from the given choices = 259129

259129 × 3 = 777387

Next highest number from the given choices = 259219

259219 × 3 = 777657

Next highest number from the given choices = 259259

259259 × 3 = 777777

Hence 259259 is the answer

138. On dividing a number by 357, 39 is obtained as remainder. On dividing the same number by 17, what will be the remainder ?A. 3 B. 4C. 5 D. 6


Answer : Option C

Explanation :

Let P be the number

Let P ÷ 357 = a, remainder = 39

Then P = 357a + 39

= (17 × 21 × a) + 34 + 5

= (17 × 21 × a) + (17 × 2) + 5

= 17(21a + 2) + 5

Hence, if the same number is divided by 17, we will get 5 as remainder.

139. 4443×6664= ?A. 22622 B. 22642C. 24622 D. 24642


Answer : Option D

Explanation :

4443×6664=444×6663×4=111×222=24642

140. A student divides a number by 5 and get 3 as remainder. If his friend divides the square of the number by 5, what will be the remainder?A. 6 B. 5C. 4 D. 3


Answer : Option C

Explanation :

(a+b)2=a2+2ab+b2

Read more...Let P be the number

P ÷ 5 = a and remainder = 3

Then P = 5a + 3

P2 = (5a + 3)2

= (5a)2 + (2 × 5a × 3) + 32

= 25a2 + 30a + 9

= 25a2 + 30a + 5 + 4

= 5(5a2 + 6a + 1) + 4

Hence when P2 is divided by 5, we will get 4 as remainder

141. What is the unit digit in 771 ?A. 4 B. 3C. 2 D. 1


Answer : Option B

Explanation :

771 = (74)17 × 73

Hence, unit digit in 771

= Unit digit in [(74)17 × 73]

= Unit digit in [(7 × 7 × 7 × 7 )17 × 73]

= Unit digit in [(9 × 9)17 × 73] (∵ 7 × 7 = 49 and 9 is the unit digit of 49)

= Unit digit in [117 × 73] (∵ 9 × 9 = 81 and 1 is the unit digit of 81)

= Unit digit in [1 × 73] (∵ 117 = 1)

= Unit digit in [73]

= Unit digit in [7 × 7 × 7]

= Unit digit in [9 × 7](∵ 7 × 7 = 49 and 9 is the unit digit of 49)


142. What is the unit digit in the product (2344 × 6892 × 349 × 527 × 238) ?A. 1 B. 2C. 3 D. 4


Answer : Option B

Explanation :

Unit digit of (2344 × 6892 × 349 × 527 × 238)

= Unit digit of (4 × 2 × 9 × 7 × 8)

= Unit digit of (8 × 9 × 7 × 8)(∵ 4 × 2 = 8)

= Unit digit of (2 × 7 × 8)(∵ 8 × 9 = 72 and 2 is the unit digit of 72)

= Unit digit of (4 × 8)(∵ 2 × 7 = 14 and 4 is the unit digit of 14)


143. Which is the common factor of (22121 + 19121) and (22231 + 19231) ?A. (22 - 19) B. (22 + 19)C. (22231 + 19231) D. (22121 + 19121)


Answer : Option B

Explanation :

(xn+an) is completely divisible by (x + a) when n is odd

Read more...Hence, (22121 + 19121) is divisible by (22 + 19) because 121 is odd

Similarly (22231 + 19231) is also divisible by (22 + 19) because 231 is odd

Hence (22 + 19) is a common factor here

144. (232+323)2−(232−323)2232×323 = ?A. 4 B. 8C. 3424 D. 2344


Answer : Option A

Explanation :

(a+b)2=a2+2ab+b2

(a−b)2=a2−2ab+b2

Read more...

Given expression is in the form (a+b)2−(a−b)2ab where a = 232 and b = 323=(a2+2ab+b2)−(a2−2ab+b2)ab=4abab=4

145. A number when divided by 44, gives 432 as quotient and 0 as remainder. What will be the remainder when dividing the same number by 31?A. 4 B. 0C. 5 D. 8


Answer : Option C

Explanation :

Let P be the number

P ÷ 44 = 432, remainder = 0

=> P = 432 × 44 + 0 = 19008

P ÷ 31 = 19008 ÷ 31 = 613, remainder = 5

Numbers

Documents