Let $S = {(m,n) : m,n \in {1,2,3,\ldots,50}}$. If the number of elements $(m,n)$ in $S$ such that $6^m + 9^n$ is a multiple of $5$ is $p$ and the number of elements $(m,n)$ in $S$ such that $m + n$ is a square of a prime number is $q$, then $p + q$ is equal to ______

$S = {1,2,3,\ldots,50}$

$p = (6^m + 9^n)$ is divisible by $5$

No. of ways

$6^m = (5\lambda + 1)^m = 5k + 1$

$9^n = (10 - 1)^n = 10\mu - 1$ if $n$ is odd

$\Rightarrow n$ must be odd

$10\mu + 1$ if $n$ is even

$\Rightarrow$ No. of ways $= 50 \times 25 = 1250$

$q \Rightarrow (m+n)$ is square of a prime

$m+n = 4, 9, 25, 49$

No. of ways: $3, 8, 24, 48$

$q = 3 + 8 + 24 + 48 = 83$

$p + q = 1250 + 83 = 1333$