Identify Prime and Composite Numbers
Some numbers, like \(72\), have many factors. Other numbers, such as \(7\), have only two factors: \(1\) and the number. A number with only two factors is called a prime number. A number with more than two factors is called a composite number. The number \(1\) is neither prime nor composite. It has only one factor, itself.
Definition: Prime Numbers and Composite Numbers
A prime number is a counting number greater than \(1\) whose only factors are \(1\) and itself.
A composite number is a counting number that is not prime.
Figure \(\PageIndex{5}\) lists the counting numbers from \(2\) through \(20\) along with their factors. The highlighted numbers are prime, since each has only two factors.
Figure \(\PageIndex{5}\): Factors of the counting numbers from 2 through 20, with prime numbers highlighted
The prime numbers less than \(20\) are \(2\), \(3\), \(5\), \(7\), \(11\), \(13\), \(17\), and \(19\). There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than \(1\) and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.
HOW TO: DETERMINE IF A NUMBER IS PRIME.
 Step 1. Test each of the primes, in order, to see if it is a factor of the number.
 Step 2. Start with \(2\) and stop when the quotient is smaller than the divisor or when a prime factor is found.
 Step 3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
Example \(\PageIndex{8}\): prime or composite
Identify each number as prime or composite:
 \(83\)
 \(77\)
Solution
 Test each prime, in order, to see if it is a factor of \(83\), starting with \(2\), as shown. We will stop when the quotient is smaller than the divisor.
Prime 
Test 
Factor of 83? 
2 
Last digit of 83 is not 0, 2, 4, 6, or 8. 
No. 
3 
8 + 3 = 11, and 11 is not divisible by 3. 
No. 
5 
The last digit of 83 is not 5 or 0. 
No. 
7 
83 ÷ 7 = 11.857…. 
No. 
11 
83 ÷ 11 = 7.545… 
No. 
We can stop when we get to \(11\) because the quotient (\(7.545…\)) is less than the divisor. We did not find any prime numbers that are factors of \(83\), so we know \(83\) is prime.
 Test each prime, in order, to see if it is a factor of \(77\).
Prime 
Test 
Factor of 77? 
2 
Last digit is not 0, 2, 4, 6, or 8. 
No. 
3 
7 + 7 = 14, and 14 is not divisible by 3. 
No. 
5 
The last digit is not 5 or 0. 
No. 
7 
77 ÷ 7 = 11 
Yes. 
Since \(77\) is divisible by \(7\), we know it is not a prime number. It is composite.
Exercise \(\PageIndex{15}\)
Identify the number as prime or composite: \(91\)
 Answer

composite
Exercise \(\PageIndex{16}\)
Identify the number as prime or composite: \(137\)
 Answer

prime
Practice Makes Perfect
Identify Multiples of Numbers
In the following exercises, list all the multiples less than 50 for the given number.
 2
 3
 4
 5
 6
 7
 8
 9
 10
 12
Use Common Divisibility Tests
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, 3, 4, 5, 6, and 10.
 84
 96
 75
 78
 168
 264
 900
 800
 896
 942
 375
 750
 350
 550
 1430
 1080
 22,335
 39,075
Find All the Factors of a Number
In the following exercises, find all the factors of the given number.
 36
 42
 60
 48
 144
 200
 588
 576
Identify Prime and Composite Numbers
In the following exercises, determine if the given number is prime or composite.
 43
 67
 39
 53
 71
 119
 481
 221
 209
 359
 667
 1771
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?