find all four digit perfect squares of the form XXYY
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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.
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.
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