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

Assume that a solution exists. It has 2 digits, x and y.    

By long multiplication, 100x^2+20xy+y^2= 1100A+11B =11(100A+B)      
So 11 divides AABB, and since AABB is a square, 11 divides it twice.      
This being the case, x=y, and  AABB = (11x)^2, where 2<x<10      
      
From the long multiplication, 21x^2 must have a doubled digit:      
      
3 189     
4 336     
5 525     
6 756     
7 1029     
8 1344     
9 1701     
      
So x=8: 88*88=7744  

Posted by broll on 2015-06-05 23:25:21