A number that is a perfect square has an odd number of factors. A number that is a 4th power has an odd number of factors. The same holds for other even-numbered powers.
The factors of 4 = 22 are 1, 2, and 4.
The factors of 9 = 32 are 1, 3, and 9.
The factors of 16 = 42 = 24 are 1, 2, 4, 8, and 16.
The factors of 25 = 52 are 1, 5, and 25.
The factors of 36 = 62 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The factors of 49 = 72 are 1, 7, and 49.
The factors of 64 = 82 = 26 are 1, 2, 4, 8, 16, 32, and 64.
The factors of 81 = 92 = 34 are 1, 3, 9, 27, and 81.
The factors of 100 = 102 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
The factors of 144 = 122 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144.
The factors of 256 = 162 = 44 = 28 are 1, 2, 4, 8, 16, 32, 64, 128, and 256.
The factors of 1024 = 322 = 210 are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024.
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Since any odd number can be a factor, there is an infinite amount of them.
A perfect square
Perfect squares are numbers with an odd number of factors.
Square numbers have an odd number of factors.
Square numbers have an odd number of factors, but beyond that, any odd number can be a factor.