Chapter 3 · Number system fluency

Prime Numbers and Prime Factorization

Sort numbers into prime and composite, find every factor pair of a number, and break a number down into a product of primes.

17 min read

Picture $12$ counters. You can set them out in equal rows in several ways: $1$ row of $12$ , $2$ rows of $6$ , or $3$ rows of $4$ . Turning a layout sideways, such as $4$ rows of $3$ , uses the same two numbers. Each layout matches a multiplication: $1 \times 12 = 12$ , $2 \times 6 = 12$ and $3 \times 4 = 12$ .

Now picture $7$ counters. The only layout with equal rows is $1$ row of $7$ , or that same layout turned sideways.

The numbers being multiplied are called factors. A factor of a whole number is a whole number that divides it exactly, with no remainder. So $3$ is a factor of $12$ , because $12 \div 3 = 4$ with nothing left over. But $5$ is not a factor of $12$ , because $12 \div 5 = 2$ remainder $2$ .

Every number in this lesson is a whole number greater than $0$ . You will sort numbers by how many factors they have, find every pair of factors a number has, and then break a number down into its smallest building blocks.

Counting factors sorts the whole numbers greater than $1$ into two groups.

A prime number is a whole number greater than $1$ that has exactly two factors: $1$ and itself. For example, $7$ is prime because its only factors are $1$ and $7$ .
A composite number is a whole number greater than $1$ that has more than two factors. For example, $12$ is composite because its factors are $1$ , $2$ , $3$ , $4$ , $6$ and $12$ .

The number $1$ is a special case. Its only factor is $1$ , so it is neither prime nor composite.

Factors of the whole numbers from 1 to 12

Number	Factors	How many factors	Prime or composite?
$1$	$1$	$1$	neither
$2$	$1$ , $2$	$2$	prime
$3$	$1$ , $3$	$2$	prime
$4$	$1$ , $2$ , $4$	$3$	composite
$5$	$1$ , $5$	$2$	prime
$6$	$1$ , $2$ , $3$ , $6$	$4$	composite
$7$	$1$ , $7$	$2$	prime
$8$	$1$ , $2$ , $4$ , $8$	$4$	composite
$9$	$1$ , $3$ , $9$	$3$	composite
$10$	$1$ , $2$ , $5$ , $10$	$4$	composite
$11$	$1$ , $11$	$2$	prime
$12$	$1$ , $2$ , $3$ , $4$ , $6$ , $12$	$6$	composite

Two rows of the table are worth a second look. The number $2$ is prime even though it is even, and $9$ is composite even though it is odd. What matters is how many factors a number has, not whether it is even or odd. In fact, $2$ is the only even prime, because every even number greater than $2$ has $2$ as a factor besides $1$ and itself.

For a larger number, you may not know the factors by heart. Instead, hunt for a factor other than $1$ and the number itself. Your divisibility rules are the quickest tools for the hunt. If you find even one such factor, the number has more than two factors, so it is composite.

Worked example

Prime or composite?

Is $51$ prime or composite?

Hunt for a factor other than 1 and 51. Start with 2. The number 51 is odd, so it is not divisible by 2. That means 2 is not a factor.
Try 3 next. Add the digits: 5 plus 1 is 6. Because 6 is divisible by 3, the number 51 is divisible by 3 too. Divide to find the number that pairs with 3.
$51 \div 3 = 17$
Division undoes multiplication, so you can rewrite the division fact as a multiplication fact. It shows that 3 and 17 are both factors of 51.
$3 \times 17 = 51$
List the factors you have found: 1, 3, 17 and 51. That is more than two factors, so 51 fits the definition of a composite number.

Answer

The number $51$ is composite, because it has the factors $3$ and $17$ as well as $1$ and $51$ .

With $51$ , the hunt ended quickly. But what if you test a number and find nothing? You need to know when it is safe to stop. Factor pairs tell you.

A factor pair of a number is two whole numbers that multiply to give that number. For example, $3$ and $4$ are a factor pair of $12$ , written $3 \times 4$ . With the counters, that was the layout with $3$ rows of $4$ . Order does not matter, so $4 \times 3$ is the same pair.

To find all the factor pairs of a number:

Test $1$ , $2$ , $3$ and so on, in order.
Each time the number you test is a factor, divide to find its partner, the other number in the pair. Write the pair down.
Stop when the number you are testing, multiplied by itself, is greater than the number you started with.

Why is it safe to stop there? Take $36$ . If both numbers in a pair were $7$ or more, they would multiply to at least $7 \times 7 = 49$ , which is more than $36$ . So every factor pair of $36$ contains a number that is $6$ or less, and you will already have tested all of those. The same reasoning works for any starting number.

Worked example

All the factor pairs of 36

Find all the factor pairs of $36$ .

Start with 1. It is a factor of every whole number, and its partner is the number itself.
$1 \times 36 = 36$
Test 2. The number 36 is even, so 2 is a factor. Divide 36 by 2 to find its partner, 18.
$2 \times 18 = 36$
Test 3. The digits 3 and 6 add to 9, which is divisible by 3, so 3 is a factor. Divide 36 by 3 to find its partner, 12.
$3 \times 12 = 36$
Test 4 and 5. Dividing 36 by 4 gives 9 with no remainder, so 4 is a factor. But 36 does not end in 0 or 5, so 5 is not a factor.
$4 \times 9 = 36$
Test 6. Dividing 36 by 6 gives 6 with no remainder, so 6 is a factor. Its partner is also 6, so this pair gives just one new factor.
$6 \times 6 = 36$
Next would be 7, but 7 times 7 is 49, which is greater than 36. Two numbers that are both 7 or more multiply to at least 49, so no new pair can exist. Stop here.
$7 \times 7 = 49 > 36$

Answer

The factor pairs of $36$ are $1 \times 36$ , $2 \times 18$ , $3 \times 12$ , $4 \times 9$ and $6 \times 6$ . So the factors of $36$ are $1$ , $2$ , $3$ , $4$ , $6$ , $9$ , $12$ , $18$ and $36$ .

The same search tells you when a number is prime. Take $29$ . Like every number, it has the pair $1 \times 29$ . Now keep testing.

$2$ is not a factor, because $29$ is odd.
$3$ is not a factor, because the digits add to $2 + 9 = 11$ , and $11$ is not divisible by $3$ .
$4$ is not a factor, because $29 \div 4 = 7$ remainder $1$ .
$5$ is not a factor, because $29$ does not end in $0$ or $5$ .
Next would be $6$ , but $6 \times 6 = 36$ , which is greater than $29$ . Stop.

The only factor pair of $29$ is $1 \times 29$ . So $29$ has exactly two factors, and it is prime. In general, a whole number greater than $1$ is prime exactly when its only factor pair is $1$ times the number itself.

A composite number can be broken down into primes. Take $12$ . One of its factor pairs is $2 \times 6$ . The $2$ is prime, but the $6$ is composite, so you can swap the $6$ for its own factor pair, $2 \times 3$ . Because $2 \times 3$ equals $6$ , the swap does not change the total. It is still $12$ .

12 = 2 \times 6 = 2 \times 2 \times 3

Now every number being multiplied is prime, so nothing can be split again. You have found the prime factorization of $12$ . The prime factorization of a number is that number written as a product of prime numbers only. A product is what you get when you multiply numbers together. A prime is written once for each time it is used, so $2$ appears twice in $2 \times 2 \times 3$ . The number $1$ never appears, because $1$ is not prime.

To find the prime factorization of any composite number, start with a factor pair that does not use $1$ . Then keep swapping each composite number for a factor pair of its own, again without using $1$ , until only primes are left. If you are not sure whether a number is prime, search for its factor pairs to find out.

Worked example

From a factor pair to primes

Write the prime factorization of $36$ .

Start with a factor pair of 36 that does not use 1. From the last example, 4 and 9 work.
$36 = 4 \times 9$
Look at the 4. It is composite, because 2 times 2 is 4. Swap the 4 for 2 times 2. You are putting in something equal to 4, so the total is still 36.
$36 = 2 \times 2 \times 9$
Look at the 9. It is composite, because 3 times 3 is 9. Swap the 9 for 3 times 3. Again the total is still 36.
$36 = 2 \times 2 \times 3 \times 3$
Check every number. 2 and 3 are both prime, so nothing can be split further. Multiply back to confirm: 2 times 2 is 4, 4 times 3 is 12, and 12 times 3 is 36.
$2 \times 2 \times 3 \times 3 = 36$

Answer

The prime factorization of $36$ is $2 \times 2 \times 3 \times 3$ .

Every starting factor pair of 36 ends with the same primes

First factor pair	Keep splitting composite numbers	Primes from smallest to largest
$2 \times 18$	$2 \times 2 \times 9$ , then $2 \times 2 \times 3 \times 3$	$2 \times 2 \times 3 \times 3$
$3 \times 12$	$3 \times 3 \times 4$ , then $3 \times 3 \times 2 \times 2$	$2 \times 2 \times 3 \times 3$
$4 \times 9$	$2 \times 2 \times 9$ , then $2 \times 2 \times 3 \times 3$	$2 \times 2 \times 3 \times 3$
$6 \times 6$	$2 \times 3 \times 6$ , then $2 \times 3 \times 2 \times 3$	$2 \times 2 \times 3 \times 3$

Key concept

Every whole number above 1 has one prime factorization

Every whole number greater than $1$ is either prime or a product of primes. However you split a composite number, you always finish with the same primes, each used the same number of times. Only the order can differ, and multiplying in a different order does not change the value. That is why you can say the prime factorization of a number. It is usual to write the primes from smallest to largest. A prime number cannot be split, so its prime factorization is just the number itself.

Worked example

Smallest prime first

Write the prime factorization of $84$ . Any way of splitting works, but here is a steady routine: keep dividing by the smallest prime that is a factor, until the result is prime.

84 is even, so the smallest prime that is a factor is 2. Divide by 2. Because division undoes multiplication, the result tells you that 84 equals 2 times 42.
$84 \div 2 = 42$
42 is even too, so divide by 2 again. This tells you that 42 equals 2 times 21, so 84 equals 2 times 2 times 21.
$42 \div 2 = 21$
21 is odd, so 2 is not a factor. Try the next prime, 3. The digits 2 and 1 add to 3, so 21 is divisible by 3. This tells you that 21 equals 3 times 7.
$21 \div 3 = 7$
7 is prime, because its only factors are 1 and 7. So stop dividing. Put the pieces together: in 2 times 2 times 21, swap the 21 for 3 times 7.
$84 = 2 \times 2 \times 3 \times 7$
Multiply back to check: 2 times 2 is 4, 4 times 3 is 12, and 12 times 7 is 84.
$2 \times 2 \times 3 \times 7 = 84$

Answer

The prime factorization of $84$ is $2 \times 2 \times 3 \times 7$ .

Here is the idea to take with you. A whole number greater than $1$ is prime when it has exactly two factors, and composite when it has more. Testing numbers in order finds every factor pair of a number. And every composite number breaks down into primes, always the same primes, however you split it.

Next, you will use these skills to find the greatest common factor and the least common multiple of two numbers.

Up next

Greatest Common Factors and Least Common Multiples

14 min read

Continue