HMMT 二月 2009 · GEN1 赛 · 第 2 题

HMMT February 2009 — GEN1 Round — Problem 2

[ 2 ] Suppose N is a 6-digit number having base-10 representation a b c d e f . If N is 6 / 7 of the number having base-10 representation d e f a b c , find N .

解析

[ 2 ] Suppose N is a 6-digit number having base-10 representation a b c d e f . If N is 6 / 7 of the number having base-10 representation d e f a b c , find N . Answer: 461538 Solution: We have 7( abcdef ) = 6( def abc ) , so 699400 a + 69940 b + 6994 c = 599300 d + 59930 e + 10 10 5993 f . We can factor this equation as 6994(100 a + 10 b + c ) = 5993(100 d + 10 e + f ), which yields 538( abc ) = 461( def ) . Since gcd(538 , 461) = 1, we must have ( abc ) = 461 and ( def ) = 538. 10 10 10 10