What Type of Number Has an Odd Number of Factors?

Only numbers that are perfect squares have an odd number of positive factors. All other types of numbers have an even number. However, if negative factors are included, then all numbers have an even number of factors.

Factors occur in pairs because pairs of factors multiplied together produce the factored number. For example, 12 is produced by multiplying 1 and 12, or 2 and 6, or 3 and 4, so 12 has six different factors. With a perfect square, there is always one factor that produces the number when multiplied by itself. For example, 16 is produced by multiplying 1 and 16, or 2 and 8, or 4 and 4, so there are only five different factors for 16.

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