12th VMO 1974 Problem A2 (1) How many positive integers n are such that n is divisible by 8 and n+1 is divisible by 25? (2) How many positive integers n are such that n is divisible by 21 and n+1 is divisible by 165? (3) Find all integers n such that n is divisible by 9, n+1 is divisible by 25 and n+2 is divisible by 4.
infinitely many, none, n = 774 mod 900 Solution (1) We need n = 0 mod 8 and -1 mod 25. Hence n = 24 mod 200. (2) We need n = 0 mod 21, n = -1 mod 165. But 3 divides 165, so we require n = 0 mod 3 and 2 mod 3, which is impossible. (3) We need n = 0 mod 9, -1 mod 25, 2 mod 4. Hence n = 99 mod 225, and 2 mod 4, so n = 774 mod 900 Thanks to Suat Namli 12th VMO 1974 © John Scholes jscholes@kalva.demon.co.uk 6 March 2004 Last corrected/updated 6 Mar 04