HMMT 二月 2007 · GEN1 赛 · 第 6 题

英文解析

6. [ 4 ] The positive integer n is such that the numbers 2 and 5 start with the same digit when writtenin 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 resultingn n n n nnumbers 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 nor 3 ≤ x, y < 4. Because n is positive, neither 2 nor 5 is a perfect power of 10, so the former isimpossible.