1st Iberoamerican 1985 Problem B2 The function f(n) is defined on the positive integers and takes non-negative integer values. It satisfies (1) f(mn) = f(m) + f(n), (2) f(n) = 0 if the last digit of n is 3, (3) f(10) = 0. Find f(1985).
If f(mn) = 0, then f(m) + f(n) = 0 (by (1)). But f(m) and f(n) are non-negative, so f(m) = f(n) = 0. Thus f(10) = 0 implies f(5) = 0. Similarly f(3573) = 0 by (2), so f(397) = 0. Hence f(1985) = f(5) + f(397) = 0. 1st Ibero 1985 © John Scholes jscholes@kalva.demon.co.uk 1 July 2002