HMMT 二月 2010 · TEAM2 赛 · 第 4 题

英文解析

4. [ 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 .
   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 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 thepossible values of a from 2 to 9, and only a = 7 yields a perfect square. When a = 7, b = 4, so theonly perfect square of for aabb is 7744.