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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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Can an odd number have odd numbers of factors?
Yes. Square numbers have an odd number of factors. 49 has three factors.
What is another name for an odd number of factors?
Square numbers have an odd number of factors.
What number has an odd number for a factor?
All odd numbers have odd factors.
Is the number of factors of any square number odd or even?
Odd.
Why are all the factors of 55 odds?
Because 55 is an odd number. All the factors of odd numbers are odd.
