How do I approach this question? I am unsure of how to use Euclid's algorithm and simultaneously incorporate that $n$ is a multiple of $58$. Thanks!
2026-02-24 02:19:12.1771899552
Show that for all $n$, If $n$ is a multiple of $58$, then $n+7$ and $n^{2}+9$ are coprime.
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If not then there exists a common divisor d > 1. It must also divide n^2+9 - n(n+7) = 9-7n. Likewise d is a divisor of 9-7n + 7(n+7) = 58. So 58 is GCD of n^2+9 and n+7.
But if n = 58m then 58m+7 cannot be a multiple of 58.