Four-digit perfect squares with repeated digits

CAT 2007 Slot 1 · QA · Medium · Number System

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Consider four-digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?

  1. A.

    3

  2. B.

    2

  3. C.

    4

  4. D.

    0

  5. E.

    1

Answer

E

Explanation

The number is N=1100a+11b=11(100a+b)N = 1100a + 11b = 11(100a + b). For NN to be a square, 100a+b100a + b must be divisible by 11   ⟹  a+b\implies a + b is divisible by 11. Testing pairs (a,b)(a, b) with a+b=11a + b = 11: 7744=8827744 = 88^2 is the only valid number.

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