A perfect square always has odd number of odd factors. A perfect square always has even number of even factors. The sum of all factors of a perfect square is always odd but if the sum of all factors of a number is odd, we cannot say that it must be a perfect square.
Therefore, the factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Here, there is an odd number of factors because the square root of the perfect square (in this case 6) does not have a pair. Therefore, perfect squares have an odd number of factors because the square root of the perfect square does not have a pair.
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