All About Programming: Prime factor - Wikipedia, the free encyclopedia

Perfect square numbers can be recognized by the fact that all of their prime factors have even multiplicities. For example, the number 144 (the square of 12) has the prime factors

144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2.

These can be rearranged to make the pattern more visible:

144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = (2 \times 2 \times 3) \times (2 \times 2 \times 3) = (2 \times 2 \times 3)^2 = (12)^2.

Because every prime factor appears an even number of times, the original number can be expressed as the square of some smaller number. In the same way, perfect cube numbers will have prime factors whose multiplicities are multiples of three, and so on.

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Prime factor - Wikipedia, the free encyclopedia

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