PUMaC 2015 · 团队赛 · 第 11 题

PUMaC 2015 — Team Round — Problem 11

[ 8 ] Given a rational number r that, when expressed in base-10, is a repeating, non-terminating decimal, we define f ( r ) to be the number of digits in the decimal representation of r that are after the decimal point but before the repeating part of r . For example, f (1 . 27) = 0 1 2 4 and f (0 . 352) = 2. What is the smallest positive integer n such that , , and are non- ( ) ( ) ( ) n n n 1 2 4 terminating decimals, where f = 3, f = 2, and f = 2? n n n 2

解析

[ 8 ] Given a rational number r that, when expressed in base-10, is a repeating, non-terminating decimal, we define f ( r ) to be the number of digits in the decimal representation of r that are after the decimal point but before the repeating part of r . For example, f (1 . 27) = 0 1 2 4 and f (0 . 352) = 2. What is the smallest positive integer n such that , , and are non- n n n ( ) ( ) ( ) 1 2 4 terminating decimals, where f = 3, f = 2, and f = 2? n n n Solution: First we figure out a nice way of thinking about f ( r ) for any given r . Let f ( r ) = k and let m be the period of the repeating part of r in base 10. Using the standard trick for converting repeating decimals to fractions, we have: k k + m k + m 10 r = integer part . repeating part 10 r = different integer part . repeating part So r (10 − integer k