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Divisibility of Mersenne numbers?

  1. Apr 23, 2010 #1
    1. The problem statement, all variables and given/known data
    Let M_n= 2^(n) - 1 be the n-th Mersenne number.
    a) Show that, if m|n, then M_m|M_n
    b) Show that, if m<n and m does not divide n, then GCD(M_n,M_m) = GCD(M_m,M_r) where r is the remainder of n upon division by m
    c) Let m,n be arbitrary natural numbers, and let d = GCD(m,n). Using the above results, show that GCD(M_n,M_n) = M_d

    2. The attempt at a solution
    So I figured out part a quite easily, by letting n=mk and using congruences, but I'm stuck on part b as im unsure whether this can by proved using only congruences or if I need to incorporate the Euclidean algorithm. Any ideas?
     
  2. jcsd
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