Smallest Exponent Sum for Perfect Cube

For $N$ to be a perfect cube, exponents must be multiples of 3.

1. $p + 7$: Smallest positive integer $p \ge 1$ giving a multiple of 3 is $p = 2$ (so $p+7 = 9$).
2. $q - 2$: Smallest positive integer $q \ge 1$ giving a multiple of 3 (or 0) is $q = 2$ (so $q-2 = 0$).
3. $r + 1$: Smallest positive integer $r \ge 1$ giving a multiple of 3 is $r = 2$ (so $r+1 = 3$).
4. $s$: Smallest positive integer $s \ge 1$ giving a multiple of 3 is $s = 3$.

Sum $p + q + r + s = 2 + 2 + 2 + 3 = 9$.