IMO 1968

Problem A2

Find all natural numbers n the product of whose decimal digits is n² - 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 (n² - 10n - 22) - n = n(n - 11) - 22 > 0 for n >= 13. So there are no solutions for n ≥ 13. But n² - 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.

10th IMO 1968

© John Scholes
jscholes@kalva.demon.co.uk
27 Sep 1998
Last corrected/updated 27 Sep 1998