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

Concept:

A number is divisible by 88 exactly when it is divisible by both 11 and 8, since 88 = 11 x 8. Divisibility by 8 depends only on the number formed by the last three digits being a multiple of 8. Divisibility by 11 depends on the difference between the sum of alternating groups of digits being 0 or a multiple of 11.

Application:

1. The number is 72X23Y, i.e. the six digits 7, 2, X, 2, 3, Y in order.

2. Test divisibility by 8 first, since it depends only on the last three digits, 23Y (the value 230 + Y).

3. Test divisibility by 11 next, using the alternating digit groups: (7 + X + 3) - (2 + 2 + Y) = 6 + X - Y, which must equal 0 or a multiple of 11.

4. Substitute X = 7 and Y = 2: the last three digits become 232, and 232 / 8 = 29, an integer, so the number is divisible by 8.

5. With X = 7 and Y = 2, 6 + X - Y = 6 + 7 - 2 = 11, which is a multiple of 11, so the number is also divisible by 11.

Cross-check:

Cross-check by forming the full number with these digits: 727232. Dividing directly, 727232 / 88 = 8264 exactly, confirming both divisibility conditions hold together.

Hence X = 7 and Y = 2.