Number Theory for GMAT.doc

7/27/2019 Number Theory for GMAT.doc

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Number Theory

Number Theory is concerned with the properties of numbers in general & in particular integers.

GMAT Number Types:

GMAT is dealing only withReal Numbers: Integers, Fractions and Irrational Numbers.Integers:

Integers are defined as: all negative natural numbers , zero , and positive natural numbers.

Irrational Numbers:

Fractions (also known as rational numbers) can be written as terminatingorrepeatingdecimals(such as 0.5, 0.76, or 0.33333...). On the other hand, all those numbers that can be written as non-terminating, non-repeating decimals are non-rational, so they are called the irrationals.Examples 2 = 1.4142356 or the number pi = ~3.14159..., from geometry. The rational & theirrationals are two totally separate number types: there is no overlap. Putting these two major

classifications, the rational numbers & the irrational, together in one set gives you the realnumbers.

Even and Odd Numbers

An even number is an integerthat is evenly divisible by 2 without a remainder. An even numberis an integer of the form n = 2k, where kis an integer.

An odd number is an integerthat is not evenly divisible by 2. An odd number is an integer of theform n = 2k+1 or n = 2k - 1, where kis an integer.

Zero is an even number.

Addition / Subtraction:even +/- even = even;even +/- odd = odd;odd +/- odd = even.

Multiplication:

even * even = even;even * odd = even;odd * odd = odd.

Division of two integers can result into an even/odd integer or a fraction.

Positive And Negative NumbersA positive number is a real number that is greater than zero.A negative number is a real number that is smaller than zero.

Zero is not positive, nor negative.

Multiplication:positive * positive = positivepositive * negative = negative


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negative * negative = positive

Division:

positive / positive = positivepositive / negative = negativenegative / negative = positive

Prime Numbers

A Prime number is a natural number with exactly two distinct natural number divisors: 1 anditself. Otherwise a number is called a composite number. Therefore, 1 is not a prime, since itonly has one divisor, namely 1. A numbern>1 is prime if it cannot be written as a product of twofactors a and b, both of which are greater than 1: n = ab.

1. The first twenty-six prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101

2. There are infinitely many prime numbers but only positive numbers can be primes.

3. The only even prime number is 2, since any larger even number is divisible by 2. Also 2is the smallest prime.4. All prime numbers except 2 and 5 end in 1, 3, 7 or 9 , since numbers ending in 0, 2, 4,

6 or 8 are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. Similarly, allprime numbers above 3 are of the form 6n-1or 6n+1, because all other numbers aredivisible by 2 or 3.

5. Any nonzero natural numbern can be factored into primes, written as a product of primesor powers of primes. Moreover, this factorization is unique except for a possiblereordering of the factors.

6. If n is a positive integer greater than 1, then there is always a prime numberp withn


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Verifying the primality (checking whether the number is a prime) of a given numbern can bedone by trial division, that is to say dividing n by all integer numbers smaller than n, therebychecking whethern is a multiple ofm 1 whose only proper factor is 1, is called a prime number. Equivalently,

one would say that a prime number is one which has exactly two factors: 1 and itself.6. Any positive factor ofn is a product of prime factors ofn raised to some power.7. If you add or subtract any two multiple ofn, the result is a multiple ofn.8. If you add or subtract a non-multiple ofn, from/to a Multiple ofn, you get a non-multiple

ofn.9. However when you add or subtract two non-multiples of 2, the result will always be

divisible by 2.10. The square of Prime Numbers have only 3 factors. Example: 22 = 4 which has only three

factors (1,2,4). 52 = 25 which has only three factors (1,5,25). 112 = 121 which has onlythree factors (1,11, 121)

11.N! is a multiple of all the integers from I toN.12. If a number equals the sum of its proper factors, it is said to be aperfect number.

Example: The proper factors of 6 are 1, 2, and 3: 1+2+3=6, hence 6 is a perfect number.

There are some elementary rules:

1. Ifa is a factorofb and a is a factor ofc, then a is a factor of(a+b). In fact, a is a factorof(ma+nb) for all integers m and n.

2. Ifa is a factor ofb and b is a factor ofc, then a is a factor ofc.3. Ifa is a factor ofb and b is a factor ofa, then a=b ora= -b.4. Ifa is a factor ofbc, andgcd(a,b)=1, then a is a factor ofc.5. Ifp is a prime number andp is a factor ofab thenp is a factor ofaorp is a factor ofb.

Finding the Number of Factors of an Integer(For Smaller Numbers Use Factor Pairs)

First make prime factorization of an integer , n=ap*bq*cr , where a, b and , c are prime factors ofn andp ,q and rare their powers. The number of factors of will be expressed by the formula(p+1)(q+1)(r+1). NOTE: this will include 1 and n itself.Example: Finding the number of all factors of 450: 450=21*32*52


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Total number of factors of 450 including 1 and 450 itself is (1+1)(2+1)(2+1)=2*3*3= 18 factors.

Finding the Sum of the Factors of an Integer

First make prime factorization of an integer , where , n=ap*bq*cr , where a, b and , c are primefactors ofn andp ,q and rare their powers.

The sum of factors of will be expressed by the formula: (ap+1

-1)*(bq+1

-1)*(cr+1

-1)(a-1)*(b-1)*(c-1)

Example: Finding the sum of all factors of 450: 450=21*32*52

The sum of all factors of 450 is: (21+1-1)*(32+1-1)*(52+1-1) = 3*26*124= 1209

(2-1)*(3-1)*(5-1) 1*2*4Greatest Common Factor (Divisor) - GCF (GCD)

The greatest common divisor (GCD), also known as the greatest common factor (GCF), orhighest common factor (HCF), of two or more non-zero integers, is the largest positive integerthat divides the numbers without a remainder.

To find the GCF, you will need to do prime-factorization. Then, multiply the common factors(pick the lowest power of the common factors).

1. Every common divisor ofa and b is a divisor of GCD (a,b).2. a*b=GCD(a,b)*lcm(a,b)3. The GCF ofa and b cannot be larger than the difference between a and b.4. Consecutive Multiples ofn has a GCF ofn.5. if and are multiples of and are units apart from each other

then is greatest common divisor of and . For example if and are

multiples of 7 and then 7 is GCD of and .

Lowest Common Multiple - LCM

The lowest common multiple or lowest common multiple (lcm) or smallest common multiple oftwo integers a and b is the smallest positive integer that is a multiple both ofa and ofb. Since itis a multiple, it can be divided by a and b without a remainder.

To find the LCM, you will need to do prime-factorization. Then multiply all the factors (pick thehighest power of the common factors).

1. If eithera orb is 0, so that there is no such positive integer, then lcm(a, b) is defined to bezero.

2. If two numbers have no Prime in common than their GFC is 1 and their LCM is simple

their product.3. The LCM of any two or more integer is always AT LEAST as large as any of the Integer.4. The LCM of TWO ODD numbers will always be ODD, and the LCM of TWO EVEN

numbers will always be EVEN.


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1. GCD and LCMThe greatest common divisor (GCD), of two or more non-zero integers, is thelargest positive integer that divides the numbers without a remainder.So GCD can only be positive integer. It should be obvious as greatest factor of

two integers can not be negative. For example if -3 is a factor of two integer then 3is also a factor of these two integers.

The lowest common multiple (LCM), of two integers and is the smallest positiveinteger that is a multiple both of and of .So LCM can only be positive integer. It's also quite obvious as if we don not limitLCM to positive integer then LCM won't make sense any more. For example whatwould be the lowest common multiple of 2 and 3 if LCM could be negative? Thereis no answer to this question.

2. DIVISIBILITY QUESTIONS ON GMATEVERY GMAT divisibility question will tell you in advance that any unknownsrepresent positive integers.

3. REMAINDERGMAT Prep definition of the remainder:If and are positive integers, there exists unique integers and , such that

and . is called a quotient and is called a remainder.Moreover many GMAT books say factor is a "positive divisor", .I've never seen GMAT question asking the remainder when dividend ( ) isnegative, but if we'll cancel this restriction (and consider ), but

leave the other restriction ( ), then division of negative integer by positiveinteger could be calculated as follow:

divided by will result: , --> , . Hence

.

TO SUMMARIZE, DON'T WORRY ABOUT NEGATIVE DIVIDENDS, DIVISORS ORREMAINDERS ON GMAT.BACK TO THE ORIGINAL QUESTION:

The integers m and p are such that 2 both and are even (as bothhave 2 as a factor) --> even divided by even can give only even remainder (0, 2, 4,

...), since remainder is not zero (as ), then remainder must be morethan 1: 2, 4, ... Sufficient.

(2) the least common multiple of m and p is 30 --> if and ,


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remainder=1=1, answer to the question would be NO. BUT if andremainder=5>1 answer to the question would be YES. Two different answers. Notsufficient.

Answer: A.

Perfect Square

A perfect square, is an integer that can be written as the square of some other integer. Forexample 16=4^2, is an perfect square. There are some tips about the perfect square:

1. The number of distinct factors of a perfect square is ALWAYS ODD. Thereverse is also true: if a number has the odd number of distinct factors thenit's a perfect square; For example 4 has distinct factor 1,2,4 (three factors). 9has distinct factors 1,2,3 (three factors). 16 has distinct factors 1,2,4,8,16(five factors).

2. The sum of distinct factors of a perfect square is ALWAYS ODD. Thereverse is NOT always true: a number may have the odd sum of its distinct

factors and not be a perfect square. For example: 2, 8, 18 or 50;3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVENnumber of Even-factors. The reverse is also true: if a number has an ODDnumber of Odd-factors, and EVEN number of Even-factors then it's a

perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor)and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverseis also true: if a number has even powers of its prime factors then it's a

perfect square. For example: , powers of prime factors 2 and 3 areeven.

Divisibility Rules

2 - If the last digit is even, the number is divisible by 2.3 - If the sum of the digits is divisible by 3, the number is also.4 - If the last two digits form a number divisible by 4, the number is also.5 - If the last digit is a 5 or a 0, the number is divisible by 5.6 - If the number is divisible by both 3 and 2, it is also divisible by 6.7 - Take the last digit, double it, and subtract it from the rest of the number, if the answer isdivisible by 7 (including 0), then the number is divisible by 7.8 - If the last three digits of a number are divisible by 8, then so is the whole number.9 - If the sum of the digits is divisible by 9, so is the number.10 - If the number ends in 0, it is divisible by 10.11 - If you sum every second digit and then subtract all other digits and the answer is: 0, or is

divisible by 11, then the number is divisible by 11.Example: to see whether 9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, thensubtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by 11, hence 9,488,699 isdivisible by 11.12 - If the number is divisible by both 3 and 4, it is also divisible by 12.25 - Numbers ending with 00, 25, 50, or 75 represent numbers divisible by 25.

Remainder


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Ifx &y arepositive integers, there exist unique integers q & r, called the quotient& remainder,respectively, such that and0r


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9. Many seemingly difficult remainder problems can be simplified using the followingformula :

Eg.

10. , can sometimes be easier calculated if we take it as

Especially when x and y are both just slightly less than n. This can be easier understoodwith an example:

Eg.

NOTE: Incase the answer comes negative, (if x is less than n but y is greater than n) thenwe have to simply add the remainder to n.

Eg. Now, since it is negative, we have to add it to

25.

11. If you take the decimal portion of the resulting number when you divide by "n", andmultiply it to "n", you will get the remainder.Note: Converse is also true. If you take the remainder of a number when divided by 'n',and divide it by 'n', it will give us the remainder in decimal format.

Eg.In this case,Therefore, the remainder is .

FactorialsFactorial of a positive integern, denoted by n!, is the product of all positive integers less than orequal to n. For instance 5!=1*2*3*4*5*.

Note: 0!=1.

Note: factorial of negative numbers is undefined.

Trailing zeros


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Trailing zeros are a sequence of 0's in the decimal representation of a number, after which noother digits follow. 125000 has 3 trailing zeros;

The number of trailing zeros in the decimal representation ofn!, the factorial of a non-negativeinteger , can be determined with this formula:

n + n + n +.....+ n, where kmust be chosen such that 5k


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Consecutive integers are integers that follow one another, without skipping any integers.7, 8, 9, and -2, -1, 0, 1, are consecutive integers.

Sum of consecutive integers equals the mean multiplied by the number of terms, n.Example: Given consecutive integers {-3,-2,-1,0,1,2}, mean= -3+2 = -1

2 2

(mean equals to the average of the first and last terms), so the sum equals to -1 *6 = -3.2 Ifn is odd, the sum of consecutive integers is always divisible by n. Given {9,10,11}, we

have n = 3 consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3. Ifn is even, the sum of consecutive integers is never divisible by n. Given {9,10,11, 12},

we have n=4 consecutive integers. The sum of 9+10+11+12=42, therefore, is notdivisible by 4.

For any set of Consecutive Integers with an ODD number of items, the sum, of all theintegers is ALWAYS a multiple of the number of items.

For any set of Consecutive Integers with an EVEN number of items, the sum, of all theintegers is NEVER a multiple of the number of items.

The product ofn consecutive integers is always divisible by n!.Example: Given n=4 consecutive integers:{3,4,5,6}. The product of 3*4*5*6 is 360,which is divisible by 4!=24.

Evenly Spaced Set

Evenly spaced set (arithmetic progression) is a sequence of numbers such that the difference ofany two successive members of the sequence is constant. The set of integers {9,13,17,21} is anexample of evenly spaced set. Consecutive integers is also an example of evenly spaced set.

If the first term is a1 and the common difference of successive members is d, then the nth

term of the sequence is given by: an = a1 + d(n - 1) In any evenly spaced set the arithmetic mean (average) is equal to the median and can

be calculated by the formula, mean = median = a1 + an2

Where a1 is the first term and an is the last term. Given set {7,11,15,19},mean=median=7 + 19 = 13

2 The sum of the elements in any evenly spaced set is given by:Sum = a1 + an * n

2 The mean multiplied by the number of terms. OR, Sum = 2a1 + d(n - 1) * n

2 The number of integer for any Evenly spaced set: (an - a1) + 1

dSpecial cases:

Sum ofn first positive integers:

Sum ofn first positive odd numbers: , where anis the


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last, nth term and given by: an = 2n - 1 . Given n = 5 first odd positive integers, then theirsum equals to .

Sum of n first positive even numbers: ,where an is the last, nth term and given by: an = 2n . Given n = 4 first positive evenintegers, then their sum equals to .

If the evenly spaced set contains odd number of elements, the mean is the middle term, sothe sum is middle term multiplied by number of terms. There are five terms in the set{1,7,13,19,25}, middle term is 13, so the sum is 13*5 =65.

Exponents

Exponents are a "shortcut" method of showing a number that was multiplied by itself severaltimes. For instance, numbera multiplied n times can be written as an, where a represents thebase, the number that is multiplied by itselfn times and n represents the exponent. The exponentindicates how many times to multiple the base, a, by itself.

Exponents one and zero:

a0 = 1 Any nonzero number to the power of 0 is 1. For example: 50 = 1 and (3)0 = 1

Note: the case of 0^0 is not tested on the GMAT.

a1 = a Any number to the power 1 is itself.

Powers of zero:

If the exponent is positive, the power of zero is zero: 0n = 0, where n>0 . If the exponent is negative, the power of zero (0n, where n


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Example: 102 = 2 = (2)2

52

3.

4. and notIf and , what is the value of ?

A.B.C.D.E.

this can be written as

Therefore,

Example: (32)4= (32) (32) (32) (32) = 3 2+2+2+2 = 38

Operations involving the same bases:

Keep the base, add or subtract the exponent (add for multiplication, subtract for division)

5.

6.Fraction as power:

7. and

8. ax+ ax+ ax = 3ax

Exponential Equations:

When solving equations with even exponents, we must consider both positive and negative

possibilities for the solutions. For instance, a2 = 25 the two possible solutions are 5 and -5.When solving equations with odd exponents, we'll have only one solution. For instance fora3= 8,solution is a = 2 and fora3 = -8, solution is a = 2.

Exponents and divisibility:

is ALWAYS divisible by .is divisible by ifn is even.


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is divisible by ifn is odd, and not divisible by a+b ifn is even.

DISTRIBUTED FORM FACTORED FORMx2 - x x(x - 1)x3- x x (x2 - 1) = x (x - 1)(x + 1)

x

4

- x

2

x

2

(x

2

- 1) = x

2

(x + 1)(x - 1)10(b + 1) 10b.1010(b - 1) 10b

1075 - 73 73(72 - 1) = 73(49 - 1) = 73. 48

58+59+510 58(1 + 5 + 52) = 58(1 + 5 + 25) = 58 .3135+35+35 35(3) = 36

ab - ab - 1 ab(1 - a-1) = ab -1(a - 1)pq + pr + qs + rs p(q + r) + s(q + r) = (p + s) (q + r)

Roots

Roots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16 andsquare root of 16=4.

General rules:

and .

, when , then and when , then . When the GMAT provides the square root sign for an even root, such as or , then the onlyaccepted answer is the positive root.

That is, 25 = 5 , NOT +5 or -5. In contrast, x2 = -x the equation has TWO solutions, +5 and -5.Even roots have only a positive value on the GMAT.

Odd roots will have the same sign as the base of the root. For example, and

.

For GMAT it's good to memorize following values:


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Last Digit Of A Product

Last n digits of a product of integers are last n digits of the product of last n digits of theseintegers. For instance last 2 digits of 845*9512*408*613 would be the last 2 digits of45*12*8*13=540*104=40*4=160=60

Example: The last digit of 85945*89*58307=5*9*7=45*7=35=5?

Last Digit Of A Power

Determining the last digit of(xyz)n:1. Last digit of (xyz)n is the same as that ofzn;2. Determine the cyclicity numberc ofz;3. Find the remainderrwhen n divided by the cyclicity;4. When r>0, then last digit of(xyz)n is the same as that ofzr and when r = 0, then last digitof(xyz)n is the same as that ofzc, where c is the cyclicity number. Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base. Integers ending with 2, 3, 7 and 8 have a cyclicity of 4. Integers ending with 4 (e.g. (xy4)n) have a cyclicity of 2. When n is odd (xy4)n will end with 4and when n is even (xy4)n will end with 6.

Integers ending with 9 (e.g. (xy9)n ) have a cyclicity of 2. When n is odd (xy9)n will end with 9and when n is even (xy9)n will end with 1.Example: What is the last digit of 12739?Solution: Last digit of 12739 is the same as 739. Now we should determine the cyclicity of:

1. 7^1=7 (last digit is 7)2. 7^2=9 (last digit is 9)3. 7^3=3 (last digit is 3)4. 7^4=1 (last digit is 1)5. 7^5=7 (last digit is 7 again!)...So, the cyclicity of 7 is 4.Now divide 39 (power) by 4 (cyclicity), remainder is 3.So, the last digit of 12739 is the same asthat of the last digit of 739, is the same as that of the last digit of 73, which is 3.


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or can be rewritten as

If from Nx - Nx-a, Nx-a is factored out we will have:

Nx-a

(Nb

- 1)Where b = x - aExample 1,58 - 55 = 55 (53 - 1) = 55 (125 - 1) = 55.124Example 2,2x - 2x-2 = 2x (22 - 1) = 2x (4 - 1) = 2x . 3

Now in Example 2 we don't know the value of the x but weknow the difference between x and x-2 is 2 therefore, b = 2.

Number Theory for GMAT.doc

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