**Problem A2**

Find all natural numbers n the product of whose decimal digits is n^{2} - 10n - 22.

**Solution**

Suppose n has m > 1 digits. Let the first digit be d. Then the product of the digits is at most d.9^{m-1} < d.10^{m-1} <= n. But (n^{2} - 10n - 22) - n = n(n - 11) - 22 > 0 for n >= 13. So there are no solutions for n ≥ 13. But n^{2} - 10n - 22 < 0 for n ≤ 11, so the only possible solution is n = 12 and indeed that is a solution.

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.

© John Scholes

jscholes@kalva.demon.co.uk

27 Sep 1998

Last corrected/updated 27 Sep 1998