We present an ancient theorem on the representation of positive integers as products of prime numbers. The consequences of this theorem are important for a host of reasons.

A factor of a number is called a *divisor*. Thus, 10 has the divisors 1, 2, 5, and 10. The divisors of *n* other than *n* are called *proper* divisors. The proper divisors of 10 are, therefore, 1, 2, and 5, while its divisors are 1, 2, 5, and 10. A divisor of *n* is said to *divide n*. Thus, 5 divides 10, while 7 does not divide 10. The symbol for “divides” is a vertical line. When *a* divides *b*, we write *a* | *b*. Whenever *a* divides *b*, observe that *b* is a *multiple* of *a*, that is, *b* = *ka*, for some integer *k*. Thus, 5 | 10, while 10 = 2 × 5. In fact, this is an “if and only if” statement:

*a*divides*b*if and only if*b*is a multiple of*a*.

In symbols, we have, using the two-way implication arrow,

Here is a nice way to think about (4.1). Even though 20 divides 100, one may write 100 = 4 × 25, preventing us from seeing the 20. On the other hand, we can write 100 = 5 × 20, thereby showing us the divisor (or factor) 20.

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