# MAT 112 Integers and Modern Applications for the Uninitiated

## Section10.1Definition 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.

### 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.
Carefully read the above again before you answer the Checkpoint 10.2 and Checkpoint 10.3 below.

### Checkpoint10.2.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

### 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.
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.