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.
Definition10.1.
An integer \(p > 1\) is prime means that the only positive factors of \(p\) are \(1\) and \(p\text{.}\) 1
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.
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
Checkpoint10.3.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.
Example10.4.Prime and composite numbers.
We give examples of prime numbers and composite numbers.
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.
Figure10.5.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.
