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:
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.