17th USAMO 1988

The repeating decimal 0.ab ... k pq ... u = m/n, where m and n are relatively prime integers, and there is at least one decimal before the repeating part. Show that n is divisible by 2 or 5 (or both). [For example, 0.01136 = 0.01136363636 ... = 1/88 and 88 is divisible by 2.]

Solution

Note that k and u are not equal (otherwise we should have regarded the repeating part as starting at k). We have m/n = ab...k/10^r pq...u/(10^r(10^s - 1) ) = (ab...k (10^s - 1) + pq...u)/(10^r(10^s - 1) ). The numerator = u - k mod 10, which is non-zero, so the numerator is not divisible by 10. But the denominator is divisible by 10. Hence after reduction to lowest terms the denominator is divisible by 2 or 5 or both.

17th USAMO 1988

© John Scholes
jscholes@kalva.demon.co.uk
21 Aug 2002