The five-digit number 47☐32 is divisible by 72. What digit goes in the box?

Explanation

1. 01

   $72 = 8 \times 9$, and 8 and 9 are co-prime, so the number must pass both tests.

2. 02

   For 9: $4 + 7 + \square + 3 + 2 = 16 + \square$ must be a multiple of 9, so $\square = 2$.

3. 03

   For 8: the last three digits become $232 = 8 \times 29$, which passes.

4. 04

   So $\square = 2$, giving $47{,}232 = 72 \times 656$.