The Fundamental Theorem of Arithmetic

Abstract

In Chapter 2, we made extensive use of the fact that every positive integer can be written in one and only one way as a product of powers of distinct primes. This property of $\mathbb Z$ is basic to mathematics. It is so basic that many people don’t even think to question it. This is especially true in school, where students spend much of elementary school working with integers, using this unique factorization property as if it were a law of nature. For example, young children build “factor trees” for whole numbers, and it is usually taken for granted that two different trees, like those in Figure 3.1, end up with the same set of prime factors.