Number & Operations for Teachers 

Copyright David & Cynthia Thomas, 2009

Composite and Prime Numbers

Modeling the Meanings of Composite and Prime

Every whole number n may be modeled as a 1 x n (i.e., one row) or n x 1 (i.e., one column) array of objects. Some of these numbers may also be modeled as rectangular arrays with more than one row or column. Figure 7.1 models four indicated products: 1 x 12, 2 x 6, 3 x 4, and 4 x 3. In each case, the written representation a x b is modeled as a rectangular array of dots having a rows, each containing b objects. In the written representation, the whole numbers a and b are called factors and the indicated product a x b is called a factorization. The indicated products 12 x 1 and 6 x 2 also represent factorizations of the whole number twelve. How would you model 12 x 1 and 6 x 2?

1 x 12

2 x 6

3 x 4

4 x 3

Figure 7.1: Four Factorizations for Twelve

If it is possible to represent a whole number n as the product of two whole numbers, neither of which is the number one, then n is called a composite number. Every composite number may be modeled as a rectangular array having more than one column or row.

Example 7.1

Name the first ten composite numbers in the set of whole numbers.

Solution 7.1

The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.

If it is impossible to represent a whole number n as the product of two whole numbers, neither of which is the number one, then n is called a prime number. Figure 7.2 models the first three prime numbers: 2, 3, and 5.

Figure 7.2: The First Three Prime Numbers

Example 7.2

Name the first ten prime numbers in the set of whole numbers.

Solution 7.2

The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

For a given composite number n, if n = a x b, where a and b are whole numbers, it is also true that a divides n (written ) and b divides n (written ). For example, because 12 = 3 x 4, we know that both 3 and 4 divide 12 evenly. In general, every composite number n is divisible by some whole number greater than one and less than itself. Why? For one reason, if n is composite, it can be modeled as a rectangular array with more than one row or column. That is, its rectangular array can be “divided” evenly into two or more rows or columns.

Every even number is composite because it is divisible by two. Likewise, all of the numbers in the set {3, 6, 9, 12, 15, …} are composite because they are divisible by three. Indeed, many whole numbers are divisible by several factors. For instance, the number 124 is divisible by 1, 2, 3, 4, 31, 62, and 124. On the other hand, the only whole numbers that divide a prime number p are one and p. For instance, the prime number 5 is only divisible by 1 and 5.

One of the oldest questions in mathematics asks whether there are an infinite number of primes. The answer to this question was known by Euclid and his contemporaries around 2300 years ago. The following argument shows why the answer to this question is yes.