Level
pending

Find a number of the form AABB which is a perfect square?

Please provide solution for it..

The answer is 7744.

**
This section requires Javascript.
**

You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.

Use modulo. Let a represent the digit A and b represent the digit B. AABB= 1100a + 11b. Square numbers are congruent to either 0 or 1 mod 4. Case 1: AABB= 0mod4. 1100a + 11b = 0mod4 1100a= 0 mod 4 so 11b=0mod4 b=4 or b=8. But a square number cannot end in 8 so b=4. So number is AA44. By checking, a=7. So 7744 is a solution.

Case 2: AABB= 1mod4. Using a similar method to above, 11b=1mod4 So Last two digits are 21, 25, 29, ...93, 97. The ones which have the same 2 digits are 33 and 77. So BB= 33 or BB=77. But a square number cannot end in 3 or 7. Thus AABB is not congruent to 1mod4.

Therefore, the only number of the form AABB which is a perfect square is 7744.