If $N=\left(7^{p+4}\right)\left(5^{q}\right)\left(2^{3}\right)$ is a perfect cube, where $p$ and $q$ are positive integers, the smallest possible value of $p+q$ is
(A) 5
(B) 2
(C) 8
(D) 6
(E) 12
If $N=\left(7^{p+4}\right)\left(5^{q}\right)\left(2^{3}\right)$ is a perfect cube, where $p$ and $q$ are positive integers, the smallest possible value of $p+q$ is
(A) 5
(B) 2
(C) 8
(D) 6
(E) 12

## Solution

In order for $N$ to be a perfect cube, each prime factor of $N$ must have an exponent that is divisible by 3. Since $p$ and $q$ must be positive integers, the smallest value of $p$ is 2 and the smallest value of $q$ is 3 . Thus, the smallest value of $p+q$ is 5 .

ANSWER: (A)