Question

For how many positive integer values of $N$ $(N \gt 0)$ is the expression

$\frac{36}{N+2}$

an integer?

$7$

$8$

$9$

$10$

$12$

$\dfrac{36}{N+2}$ is an integer exactly when $N+2$ is a divisor of $36$

Since $N \gt 0,$ only divisors of $36$ that are greater than $2$ count

The expression is an integer when $N+2$ divides $36.$ The divisors of $36$ are $1,$ $2,$ $3,$ $4,$ $6,$ $9,$ $12,$ $18,$ $36.$

Because $N \gt 0,$ we need $N + 2 \gt 2,$ leaving the divisors $3,$ $4,$ $6,$ $9,$ $12,$ $18,$ $36$ : that is $7$ values.

Thus, the correct answer is A .