If x and y are the two digits the number 653xy such that this number is…

Divisibility concept: to test divisibility by a composite number that splits into two coprime factors (gcd = 1), it is enough to check divisibility by each factor separately, and that check is exact. 80 = 16 × 5, and gcd(16, 5) = 1, so a number is divisible by 80 if and only if it is divisible by both 16 and 5. (Splitting 80 as 8 × 10 instead is unsafe, because gcd(8, 10) = 2, so passing both those tests only guarantees divisibility by their LCM, 40, not by 80.)

1. Divisibility by 5 requires the units digit to be 0 or 5.

2. Divisibility by 16 requires the number to be even, so the units digit cannot be 5; hence y = 0.

3. Divisibility by 16 depends on the last four digits of the number. With y = 0, the last four digits form the number 5300 + 10x.

4. 5300 leaves remainder 4 on dividing by 16 (5300 = 16 × 331 + 4), so we need 10x to leave remainder 12 on dividing by 16.

5. Checking x = 0 to 9, only x = 6 gives 10x = 60, which leaves remainder 12 on dividing by 16 (60 = 16 × 3 + 12). So x = 6.

6. Therefore x + y = 6 + 0 = 6.

Verify directly: with x = 6, y = 0 the number is 65360, and 65360 ÷ 80 = 817 exactly, confirming divisibility by 80. By contrast, testing only 8 and 10 (the flawed shortcut) would also pass x = 2, since 65320 is divisible by both 8 and 10; but 65320 ÷ 80 = 816.5, so it is NOT divisible by 80 — showing why the 16-and-5 test must be used instead.