$33 \quad m$ and $n$ are two positive integers satisfying $1 \leq m \leq n \leq 40$. Find the number of pairs of $(m, n)$ such that their product $m n$ is divisible by 33 .
33 Answer: (64)
If $n=33, m=1,2,3, \ldots, 32$, so there are 32 pairs.
If $m=33, n=33,34, \ldots, 40$, so there are 8 pairs.
If $n, m \neq 33$, we have 2 cases:
Case 1: $m=3 a, n=11 b(a \neq 11$ and $b \neq 3)$, thus $1 \leq 3 a \leq 11 b \leq 40$. So if $b=1, a=1,2,3$ and if $b=2, a=1,2,3, \ldots, 7$. There are 10 pairs.
Case 2: $m=11 a, n=3 b(a \neq 3$ and $b \neq 11)$, thus $1 \leq 11 a \leq 3 b \leq 40$.
So if $a=1, b=4,5,6,7,8,9,10,12,13$ and if $a=2, b=8,9,10,12,13$.
There are $9+5=14$ pairs.
Hence we have altogether $32+8+10+14=64$ pairs.