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composite number

A positive integer that can be factored into smaller positive integers, neither of which is one. If a positive integer is not composite (4, 6, 8, 9, 10, 12, ...) or one, then it is a prime number (2, 3, 5, 7, 11, 13, 17, ...). As Karl Gauss put it in his Disquisitiones Arithmeticae (1801): "The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic." One reason it is important today is that many secret codes and much of the security of the Internet depends in part on the relative difficulty of factoring large numbers. But more basic to a mathematician is that this problem has always been central to number theory.

Numbers that, for their size, have a lot of factors are sometimes referred to as highly composite numbers. Examples include 12, 24, 36, 48, 60, and 120.

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