n is a positive integer that has exactly 3 prime factors, 2, 3 and 5.

Question

n is a positive integer that has exactly 3 prime factors, 2, 3 and 5. What is the value of n?

(1) n has a total of 12 positive factors, including 1 and n 
(2) n > 100

SajjitaKundu · Answer

IMO the answer is C. We have to find powers of 2 3 and 5 for this number.

amanvermagmat · Answer

A positive integer with prime factorisation as: p^a * q^b * r^c.... (where p, q, r.. are distinct prime numbers and a, b, c are positive integers) has its total number of factors given by = (a+1)*(b+1)*(c+1)..

Here n is a positive integer with exactly 3 prime factors. So n will be = 2^a * 3^b * 5^c, and a/b/c will be positive integers. Number of factors of n will be = (a+1)*(b+1)*(c+1). We have to find the value of n.

(1) Given that (a+1)*(b+1)*(c+1) = 12. This is only possible when out of a, b, c - two integers have a value of 1 each and one integer has a value of 2. Because (2+1)*(1+1)*(1+1) = 3*2*2 = 12. So n can thus take following values:
Either 2^2 * 3 * 5 = 60
OR 2 * 3^2 * 5 = 90
OR 2 * 3 * 5^2 = 150
Since there is not a unique value, this statement is not sufficient.

(2) n > 100. Obviously this is not sufficient.

Combining the two statements, if n can be either 60 or 90 or 150 but n has to be > 100, then n can only be = 150. Sufficient.

Hence  C answer

n is a positive integer that has exactly 3 prime factors, 2, 3 and 5.

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