Prime Numbers

This section is all about basic properties of positive whole numbers, especially with regard to multiplication. Most of the time, when we say ``number'' we mean positive integer. Here is some terminology:

Multiple:

We say one number a is a multiple of another b if there is a positive integer c for which a=bc.
Example: 35 is a multiple of 7 because $35=7\times 5$.

Divides:

We say one number a divides another b if there is a positive integer c for which b=ac. In other words, b is a multiple of a.
Example: 7 divides 35 because $35=7\times 5$.
We also say b is divisible by a, and we write a|b (pronounced ``a divides b''). We can also call a a factor or divisor of b.

Factorization:

A factorization of a number a is a way of writing a as a product of smaller numbers. For instance, $6\cdot 5\cdot 8$ is a factorization of 240.

Prime:

A number p is prime if it is bigger than 1 and its only divisors are 1 and itself.

Composite:

A number is composite if it is bigger than 1 and not prime. That means it has divisors other than itself and 1.

Prime Factorization:

A prime factorization of a number is a way of writing it as a product of prime numbers. For instance, $2\cdot 3\cdot2 \cdot 5 \cdot 2 \cdot 2$ isa prime factorization of 240.

Greatest Common Divisor:

The greatest common divisor of two numbers a and b is the largest number d that divides both a and b. For example, the greatest common divisor of 30 and 42 is 6. The greatest common divisor is also sometimes called the greatest common factor or highest common factor.

Relatively Prime:

Two numbers a and b are relatively prime if their greatest common divisor is 1.

A proof that there are infinitely many prime numbers appeared in Euclid's Elements more than 2000 years ago.

The Fundamental Theorem of Arithmetic states that every number has a unique prime factorization, subject to rearrangement of the prime factors. For example, the standard way of writing the prime factorization of 240 is

$$240 = 2^4 \cdot 3 \cdot 5$$

For an introduction to both of these theorems, please read Chapters 1 and 3 of Constance Reid's From Zero To Infinity! [Reid, 1992].

Click here to see some of From Zero to Infinity!