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In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly.〔 〕 The prime factorization of a positive integer is a list of the integer's prime factors, together with their multiplicities; the process of determining these factors is called integer factorization. The fundamental theorem of arithmetic says that every positive integer has a single unique prime factorization. To shorten prime factorizations, factors are often expressed in powers (multiplicities). For example, : in which the factors 2, 3 and 5 have multiplicities of 3, 2 and 1, respectively. For a prime factor ''p'' of ''n'', the multiplicity of ''p'' is the largest exponent ''a'' for which ''pa'' divides ''n'' exactly. For a positive integer ''n'', the ''number'' of prime factors of ''n'' and the ''sum'' of the prime factors of ''n'' (not counting multiplicity) are examples of arithmetic functions of ''n'' that are additive but not completely additive. ==Perfect squares== 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 : These can be rearranged to make the pattern more visible: : 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Prime factor」の詳細全文を読む スポンサード リンク
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