Question

Solução da A1 do IMO-1964​

Answer 1

23 = 1 (mod 7). Hence 23m = 1 (mod 7), 23m+1 = 2 (mod 7), and 23m+2 = 4 (mod 7). Hence we never have 7 dividing 2n + 1, and 7 divides 2n - 1 iff 3 divides n.

Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.