BMO 1967 Problem 8 For which positive integers n does 61 divide 5 n - 4 n ? Solution
: multiples of 3. Clearly 5 - 4 = 1 (mod 61), 5 2 - 4 2 = 9 (mod 61) and 5 3 - 4 3 = 0 (mod 61). So it follows that 5 3m = 4 3m (mod 61). In other words 61 does divide 5 n - 4 n if n is a multiple of 3. It remains to show that it does not divide 5 n - 4 n if n = 3m + 1 or 3m + 2. 5 3m+1 = 5 5 3m = 5 4 3m = 4 3m+1 + 4 3m (mod 61). But 61 does not divide 4 3m , and hence it does not divide 5 3m+1 - 4 3m+1 . Similarly, 5 3m+2 = 25 5 3m = 25 4 3m = 4 3m+2 + 9 4 3m (mod 61). But 61 does not divide 9 or 4 3m , so it does not divide 5 3m+2 - 4 3m+2 . 3rd BMO 1967 © John Scholes jscholes@kalva.demon.co.uk 30 Aug 2000