The largest integer p for which $p + 10$ $p + 10$ divides $p^{3} + 100$ $p^3 + 100$ ?

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Answer 1 · 2015-10-12T19:44:35

The answer is $890$ $890$ .

This is an interesting question.

$p^{3} + 100 = (p + 10) (p^{2} - 10 p + 100) - 900$ $p^3 + 100 = (p+10)(p^2-10p+100) - 900$

So if $p + 10$ $p+10$ is a divisor of $p^{3} + 100$ $p^3+100$ , then it must also be a divisor of $- 900$ $-900$ .

The largest integer divisor of $- 900$ $-900$ is $900$ $900$ , yielding $p = 890$ $p = 890$ .

The largest integer p for whichp+10 p + 10 divides p3+100p^3 + 100?