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This problem trains for: SAT-I, AMC-8, GMAT.
A square with a side of integer length has the same area as a rectangle with dimensions:
The area of a rectangle is:
We must choose the rectangle that has an area equal to a perfect square:
Only the first choice of numbers works out to be a perfect square: 1764.
To solve this quickly, do not multiply each example and take the square root. Instead, factor all into primes and check that the product has only even powers for each prime. This is how our winning combination looks like:
and this is one of the losers:
All this factoring is easy enough to do mentally and the problem should not require the use of a calculator, or even any figuring on paper.