No or Yes?

Is the following statement true?

                          A perfect square (excluding 0) always has an odd number of factors.
No Yes

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2 solutions

Henry U
Oct 26, 2018

Factors usually come in pairs, every factor has a corresponding second factor so that their product is the factorized number. With perfect squares, however, the square root is its own corresponding factor. But since all factors are only counted once, every perfect square has an odd number of factors

Nashita Rahman
Oct 27, 2018

A positive perfect square will always be raised to an even power so the number of its factors will be even + 1 ( because one is also a factor) . We know, that an even number plus 1 is always odd . Hence, the number of factors for a perfect square is odd.

For example, let n be a prime number where n 2 n^{2} = n x n x 1 (there are three factors that is odd number of factors) .

Note: In case where n is not prime, the factors of n 2 n^{2} will always be raised to an even power and one is also a factor so the number of factors in total will be odd.

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