## Ask Professor Puzzler

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Bramasta asks, "Is it possible to have a number that is a multiple of 2 but not a multiple of 4 and is a perfect square number?"

That's a great question, Bramasta, and I'm guessing you've tried a few examples, and decided the answer is probably "no," but you're wondering if we can know it for sure.

Another way of describing "a number that is a multiple of 2 but not a multiple of 4" is to say "a number which is two more than a multiple of four." So if we can show that no perfect squares are two more than a multiple of four, we'll have answered your question. Ready?

If Y is a perfect square, that means its square root, X, is an integer. Consider that X must be either even or odd. In other words, for some value K,

X = 2K or

X = 2K + 1

Since Y = X^{2}, Y = (2K)^{2} or Y = (2K + 1)^{2}. Multiplying these out gives us the following:

Y = 4K^{2} or Y = 4K^{2} + 4K + 1 = 4(K^2 + K) + 1

Notice that in the first case, Y is a multiple of 4, and in the second case, it's one more than a multiple of four. In no case do we get a result that is either 2 more than a multiple of 4 or 3 more than a multiple of four. So we've actually proved MORE than you asked; if a number is either 2 more or 3 more than a multiple of 4, we can emphatically declare that it is NOT a perfect square.

Thanks for asking!