What are the values of X & Y in 72X23Y for it to be perfectly divisible by 88?

Concept: 88 = 8 × 11, and 8 and 11 share no common factor, so a number is divisible by 88 exactly when it is divisible by BOTH 8 and 11. Rule for 11: taking the digits from the left, compute (sum of digits at odd positions) − (sum of digits at even positions); the number is divisible by 11 only if this difference is 0 or a multiple of 11. Rule for 8: a number is divisible by 8 only if the 3-digit number formed by its last three digits is divisible by 8.

Application: For 72X23Y, the digits in order are 7, 2, X, 2, 3, Y. Taking positions from the left, the odd-position digits are 7, X, 3 and the even-position digits are 2, 2, Y, so the rule for 11 requires (7 + X + 3) − (2 + 2 + Y), i.e. (X − Y + 6), to be 0 or a multiple of 11. Testing each candidate pair:

1. X = 1, Y = 5: X − Y + 6 = 1 − 5 + 6 = 2 (not a multiple of 11).

2. X = 7, Y = 5: X − Y + 6 = 7 − 5 + 6 = 8 (not a multiple of 11).

3. X = 3, Y = 2: X − Y + 6 = 3 − 2 + 6 = 7 (not a multiple of 11).

4. X = 7, Y = 2: X − Y + 6 = 7 − 2 + 6 = 11 (a multiple of 11) — only this pair satisfies the rule for 11.

Cross-check: With Y = 2, the last three digits of 72X23Y are 232. Since 232 ÷ 8 = 29 exactly, the number is also divisible by 8. X = 7 and Y = 2 therefore satisfy both conditions required for divisibility by 88.

Hence, X = 7 and Y = 2 is the correct pair.