Discussion
The number 57x63y is divisible by 72. What is the value of \(x+y\)?
Answer: C
Correct answer: 6
Since \(72 = 8 \times 9\) and 8 and 9 share no factor, both tests must pass.
Divisibility by 8. Only the last three digits matter: 63y, that is \(630 + y\). Because \(630 = 8\times 78 + 6\), we need \(6 + y \equiv 0 \pmod 8\), and the only digit that works is \(y = 2\).
Divisibility by 9. The digit sum must be a multiple of 9: \(5+7+x+6+3+2 = 23 + x\). The only digit giving a multiple of 9 is \(x = 4\), for a sum of 27.
The number is 574632, and \(574632 \div 72 = 7981\). Hence \(x + y = 4 + 2 = 6\).
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