HMMT 二月 2007 · GEN1 赛 · 第 6 题

HMMT February 2007 — GEN1 Round — Problem 6

[ 4 ] The positive integer n is such that the numbers 2 and 5 start with the same digit when written in decimal notation; determine this common leading digit.

解析

[ 4 ] The positive integer n is such that the numbers 2 and 5 start with the same digit when written in decimal notation; determine this common leading digit. 2 2 2 2 2 2 n n Answer: 3 . Note 1 = 1 < 2 < 3 < 10 < 4 < · · · < 9 < 10 = 100. Divide 2 and 5 by 10 repeatedly until each is reduced to a decimal number less than 10 but at least 1; call the resulting n n n n n numbers x and y . Since (5 )(2 ) = 10 , either xy = 1 or xy = 10. Because 2 and 5 begin with the same digit, x and y are bounded by the same pair of adjacent integers. It follows that either x = y = 1 n n or 3 ≤ x, y < 4. Because n is positive, neither 2 nor 5 is a perfect power of 10, so the former is impossible.