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 ?  Select:
A) 3 B) 2 C) 4 D) 0 E) 1
Explanation :   If we let the four-digit number be XXYY, then this number can be expressed as:   1000X + 100X + 10Y + Y = 1100X + 11Y = 11(100X + Y) = k^2 (since it's a perfect square)   In order for this to be true, 100X + Y must be the product of 11 and a perfect square, and looks like X0Y. So now our question is "which product of 11 and a perfect square looks like X0Y?" We can test them:   11 x 16 = 176; 11 x 25 = 275; 11 x 36 = 396; 11 x 49 = 593; 11 x 64 = 704; 11 x 81 = 891   The only one that fits the bill is 704.   This means there is only one four-digit number that works, and it's 7744.   Answer : 1
<result>E</result>
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 ?