What are the values for X and Y in 72X23Y for it to be perfectly divisible by…

Solution: We need 72X23Y to be divisible by 88, so it must be divisible by both 8 and 11.

• Divisible by 8: Check the last three digits, which are 23Y. 230 mod 8 = 6, so we need Y such that 6 + Y ≡ 0 (mod 8). That gives Y = 2 (since 6 + 2 = 8). Thus Y = 2.

• Divisible by 11: Use the alternating-sum rule. Sum of digits in odd positions (from left) = 7 + X + 3 = 10 + X. Sum of digits in even positions = 2 + 2 + 2 = 6. The difference is (10 + X) − 6 = 4 + X, which must be a multiple of 11. The only digit X (0–9) that makes 4 + X = 11 is X = 7.

Conclusion: X = 7 and Y = 2, so the number 727232 is divisible by 88.