HMMT 二月 2010 · TEAM2 赛 · 第 4 题

HMMT February 2010 — TEAM2 Round — Problem 4

[ 20 ] Find all 4-digit integers of the form aabb (when written in base 10) that are perfect squares.

解析

[ 20 ] Find all 4-digit integers of the form aabb (when written in base 10) that are perfect squares. 2 2 Answer: 7744 Let x be an integer such that x is of the desired form. Then 1100 a + 11 b = x . 2 Then x is divisible by 11, which means x is divisible by 11. Then for some integer, y , x = 11 y . Then 2 2 2 1100 a +11 b = 11 y ⇒ 100 a + b = 11 y . This means that 100 a + b ≡ 0 (mod 11) ⇒ a + b ≡ 0 (mod 11). Because a and b must be nonzero digits, we have 2 ≤ a, b ≤ 9, so we can write b = 11 − a . 2 2 Replacing b in the equation derived above, we obtain 99 a + 11 = 11 y ⇒ 9 a + 1 = y . We check the possible values of a from 2 to 9, and only a = 7 yields a perfect square. When a = 7, b = 4, so the only perfect square of for aabb is 7744.