ABSTRACT
The natural numbers and integers differ from the real numbers in having "atoms" that cannot be "split" into products. To discuss primes mathematicians need to be clear about the concept of division in the positive integers. This involves a return to the concept of division learned in school: division with remainder. The idea of displaying multiples of an integer as equally spaced points along the number line gives a striking insight into thegreatest common divisor of integers a and b, the greatest integer that divides both a and b. There are many ways to use unique prime factorization, and it is rightly regarded as a powerful idea in number theory In fact, it is more powerful than Euclid could have imagined. There are complex numbers that behave like "integers" and "primes," and unique prime factorization holds for them as well.
