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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