I get some troubles with this problem, could you help me?
Given prime numbers $p$ and $q$ prove $\exists k\in\mathbb{Z}$ such that $pn^q+qn^p+kn$ is a multiple of $pq,~\forall n\in\mathbb{Z}$.
A Divisibility Problem with Prime Numbers
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$\begingroup$
number-theory
prime-numbers
divisibility
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0What about using the CRT? – 2017-01-10
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3CRT not really needed. Just Fermat's little Theorem. But note, the problem needs $p,q$ to be _distinct_ primes (else the problem is wrong). First consider the expression mod $p$, then mod $q$. A valid $k$ will then be obvious. – 2017-01-10
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0Sorry -- my previous comment is wrong -- you do need CRT. Also, $p,q$ need not be distinct. But my other hints are still OK. – 2017-01-10
1 Answers
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Hint $\ $ Write it as $\ p(n^{\large q}\!-n)+q(n^{\large p}\!-n) +\!\! \overbrace{(p\!+\!q+k)}^{\large =\ 0\ \ {\rm if}\ \ k\ =\ \ldots}\!\!\!n\ $ and apply little Fermat