AABB ....

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

Please provide solution for it..


The answer is 7744.

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1 solution

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.

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