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Perfect squares have an odd number of factors.

If f is a factor of N, then so is N/f. (For example, 3 is a factor of 30, and so is 10). Thus, it seems factors come in pairs, and there should always be an even number of them. But if f = N/f, then these two factors are the same, resulting in an odd number of factors. For example, 3 is a factor of 9, and it's "mate" is also 3.

So for the numbers with an odd number of factors, there is some f where f=N/f. Multiplying both sides by f, we have f^2 = N. So this happens when N is a perfect square.

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12y ago

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Q: What is an odd number of factors?
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