A number AABB is the square of an integer. Find this integer, aided by pen and paper. No other calculating aids allowed.

A number in the form AABB is divisible by 11, so it is 11 x A0B, where that middle digit is a zero.

   To check whether a number is divisible by 11, sum the digits in the  odd positions counting from the left (the first, third, ....) and  then sum the remaining digits.
   If the difference between the two sums is divisible by 11,    then so is the original number. Otherwise it is not.

So for A0B to be divisible by 11 (necessary to match the other factor of 11 to make a square), A+B must be divisible by 11, which means it must be 11, since the most it could be is 9+9=18.

Of the possibilities, 209, 308, ..., 803 and 902, only 704, when divided by 11, leaves another perfect square (64) so that the whole number would be a perfect square: 704*11=7744.

Posted by Charlie on 2004-06-25 09:55:56