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Section 10.1 Definition of a Prime

Recall that for two integers \(a\) and \(b\text{,}\) \(b\) divides \(a\) means that \(b\) is a factor of \(a\) (see Definition 4.1). Every positive integer, \(n\text{,}\) has the property that both \(1\) and \(n\) are factors of \(n\text{.}\) Prime numbers have only these factors.

Definition 10.1.

An integer \(p > 1\) is prime means that the only positive factors of \(p\) are \(1\) and \(p\text{.}\)
For a history of the choice not to consider the number 1 a prime number see What is the Smallest Prime ?[1]
An integer greater than 1 that is not prime is called composite.
As a prime number is only divisible by 1 and itself, a composite number \(n\) has at least one other factor \(a\) (that is not \(1\) or \(n\)).
So we can say that an integer \(n>1\) is composite if it can be written as \(n=a\cdot b\text{,}\) where \(a\) and \(b\) are integers greater than 1.
Carefully read the above again before you answer the Checkpoint 10.2 and Checkpoint 10.3 below.

Checkpoint 10.2. Prime numbers.

Let n be a natural number greater than 1. If n and 1 are the only positive factors of n, then (select all statements that are true):
  • n has a positive factor a with a \(\ne\) 1 and a \(\ne\) n
  • n is prime.
  • n is not prime.
  • n is composite.
  • n is not composite

Checkpoint 10.3. Composite numbers.

Let n be a natural number. If there are natural numbers a and b with a\(\ne\)1 and b\(\ne\)1 such that n=a\(\cdot\)b, then (select all statements that are true):
  • The positive factors of n include 1, a, b and n.
  • n is not composite
  • n is composite.
  • n is prime.
  • n is not prime.
  • The only positive factors of n are 1 and n.
We list the first \(11\) prime and composite numbers along with a factorization.

Example 10.4. Prime and composite numbers.

We give examples of prime numbers and composite numbers.
  1. The first \(11\) primes numbers are
    \begin{equation*} 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 \end{equation*}
  2. The first \(11\) composite numbers along with a representation as a product are
    \begin{gather*} 4=2\cdot 2\\ 6=2\cdot 3\\ 8=2\cdot 4\\ 9=3\cdot 3\\ 10=2\cdot 5\\ 12=2\cdot 6\\ 14=2\cdot 7\\ 15=3\cdot 5\\ 16=4\cdot 4\\ 18=3\cdot 6\\ 20=2\cdot 10\text{.} \end{gather*}
    The representation of composite numbers as products are not always unique, for example we have \(12=2\cdot 6\) and also \(12=3\cdot 4\text{.}\)
In the video in Figure 10.5 we summarize the material above.
Figure 10.5. Definition of a Prime by Matt Farmer and Stephen Steward. Early in the video we write \(a\mid b\) for \(a\) divides \(b\) where \(a\) and \(b\) are integers.