GMAT Probability and Number Theory Problem Solving Question

GMAT QUANTITATIVE REASONING

NUMBER PROPERTIES &

PROBABILITY

PROBLEM SOLVING

QUESTION 4

Q-51 Series

Question

If two distinct integers a and b are picked from {1, 2, 3, 4, .... 100} and multiplied, what

is the probability that the resulting number has EXACTLY 3 factors?

A.4

25×99

B.2

25×99

C.8

25×99

D.16

25×99

E.32

25×99

◴What kind of numbers have 3 factors?

Part 1

If two distinct integers a and b are picked from {1, 2, 3, 4, .... 100} and multiplied, what is the probability that

the resulting number has EXACTLY 3 factors?

What type of numbers have 3 factors?

1

Let n = a * b




1

Let n = a * b

1 & n are two

factors of n




1

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n




1 2

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n

Because n has

exactly 3 factors




1 2

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n

n has to be of

the form x * x

Because n has

exactly 3 factors




1 2

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n

n has to be of

the form x * x

Because n has

exactly 3 factors

So, n is a

perfect square




1 2 3

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n

n has to be of

the form x * x

Because n has

exactly 3 factors

So, n is a

perfect square

What if x had p &

q as its factors?




1 2 3

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n

n has to be of

the form x * x

Because n has

exactly 3 factors

So, n is a

perfect square

What if x had p &

q as its factors?

p & q will also

be factors of n




1 2 3

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n

n has to be of

the form x * x

Because n has

exactly 3 factors

So, n is a

perfect square

What if x had p &

q as its factors?

p & q will also

be factors of n

Then n will have

more than 3 factors




1 2 3 4

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n

n has to be of

the form x * x

Because n has

exactly 3 factors

So, n is a

perfect square

What if x had p &

q as its factors?

p & q will also

be factors of n

Then n will have

more than 3 factors

If n has to have

exactly 3

factors, x cannot

have factors

other than 1 & x.




1 2 3 4

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n

n has to be of

the form x * x

Because n has

exactly 3 factors

So, n is a

perfect square

What if x had p &

q as its factors?

p & q will also

be factors of n

Then n will have

more than 3 factors

If n has to have

exactly 3

factors, x cannot

have factors

other than 1 & x.

i.e., x is prime




1 2 3 4

Let n = a * b

1 & n are two

factors of n

Let x be the

3rd factor of n

n has to be of

the form x * x

Because n has

exactly 3 factors

So, n is a

perfect square

What if x had p &

q as its factors?

p & q will also

be factors of n

Then n will have

more than 3 factors

If n has to have

exactly 3

factors, x cannot

have factors

other than 1 & x.

i.e., x is prime

‘n’ is the square of a prime number



How many such numbers exist from 1 to 100?

1 n = a * b, where both a and b are distinct





2 n has 3 factors; 1, n and x.So, n = x * x or n = 1 * n






3a and b have to therefore be 1 and n as aand b are distinct







4 So, n is the square of a prime number andlies between 1 and 100, inclusive.








List of possible values that n can take

22 = 4









22 = 4

32 = 9









22 = 4

32 = 9

52 = 25









22 = 4

32 = 9

52 = 2572 = 49




n can take

4 values






22 = 4

32 = 9

52 = 2572 = 49

◴Finding the required probability

Part 2



Expression to find the required probability

Probability = Total number of outcomes

Favorable outcomes



Expression to find the required probability

Probability = Total number of outcomes

Favorable outcomes

Probability = Number of outcomes in which the product of the selected two has exactly 3 factors

Number of ways of selecting two distint integers from 100



Number of ways of selecting two integers : Denominator

= 100C2Number of ways of selecting two distinct

positive integers from {1, 2, 3, …., 99, 100}

Denominator



Number of ways of selecting two integers : Denominator

= 100C2Number of ways of selecting two distinct

positive integers from {1, 2, 3, …., 99, 100}

Denominator

100C2 = 100×99

1×2= 50 99



Ways to select the favorable outcomes : Numerator

Number of ways in which the product of the

numbers selected has exactly 3 factors






Values that n (= a b) can take are 4, 9, 25, 49






Values that n (= a x b) can take are 4, 9, 25, 49

The number of ways of expressing 4 as a

product of two distinct positive integers is only

ONE i.e., 1 x 4









ONE i.e., 1 x 4

The same holds good for the other 3 numbers

as well. Each has only ONE way.









ONE i.e., 1 x 4

The same holds good for the other 3 numbers

as well. Each has only ONE way.

NUMERATOR

Total number of ways= 4



The Probability

Required Probability = 4

50 × 99



The Probability


50 × 99

= 2

25 × 99



The Probability


50 × 99

= 2

25 × 99

Choice B is the answer

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GMAT Probability and Number Theory Problem Solving Question

Education

xbecause n

factors of nif

factor of nbecause n

form x

distinct integers

resulting number

type of numbers

factor of nn