GMAT Probability and Number Theory Problem Solving Question
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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
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 & n are two
factors of n
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 & n are two
factors of n
Let x be the
3rd factor of n
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 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
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 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
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 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
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 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?
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 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
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 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
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 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.
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 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
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 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
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?
How many such numbers exist from 1 to 100?
1 n = a * b, where both a and b are distinct
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?
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
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?
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
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?
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.
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?
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
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?
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
32 = 9
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?
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
32 = 9
52 = 25
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?
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
32 = 9
52 = 2572 = 49
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?
How many such numbers exist from 1 to 100?
n can take
4 values
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
32 = 9
52 = 2572 = 49
◴Finding the required probability
Part 2
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?
Expression to find the required probability
Probability = Total number of outcomes
Favorable outcomes
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?
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
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?
Number of ways of selecting two integers : Denominator
= 100C2Number of ways of selecting two distinct
positive integers from {1, 2, 3, …., 99, 100}
Denominator
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?
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
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?
Ways to select the favorable outcomes : Numerator
Number of ways in which the product of the
numbers selected has exactly 3 factors
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?
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
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?
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 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
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?
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 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
The same holds good for the other 3 numbers
as well. Each has only ONE way.
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?
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 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
The same holds good for the other 3 numbers
as well. Each has only ONE way.
NUMERATOR
Total number of ways= 4
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?
The Probability
Required Probability = 4
50 × 99
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?
The Probability
Required Probability = 4
50 × 99
= 2
25 × 99
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?
The Probability
Required Probability = 4
50 × 99
= 2
25 × 99
Choice B is the answer
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