# More practice problems for RMO on Divisibility

1. Show that the number $1^{47}+2^{47}+3^{47}+4^{47}+5^{47}+6^{47}$ is a multiple of 7.
2. Compute the value of the expression $\frac{(10^{4}+324)(22^{4}+324)(34^{4}+324)(46^{4}+324)(58^{4}+324)}{(4^{4}+324)(16^{4}+324)(28^{4}+324)(40^{4}+324)(52^{4}+324)}$.
3. Given $s+1$ integers $a_{0}, a_{1}, \ldots, a_{s}$ and a prime number p, show that p divides the integer $N(n)=a_{0}+a_{1}n +\ldots + a_{s-1}n^{s-1}+a_{s}n^{s}$ if and only if p divides $N(r)$ for an integer r, $0 \leq r \leq p-1$. Use this to find all integers n such that 7 divides $3n^{2}+6n+5$.
4. Show that there exist infinitely many pairs of integers $\{ x,y\}$ satisfying $x+y=40$ and $(x,y)=5$.
5. Prove that one cannot find integers m and n such that $m+n=101$ and $(m,n)=3$.

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