Represent composite numbers with prime factor trees

Ontario Grade 6 Mathematics

Number

Split composite numbers into factors until every branch ends in a prime

A factor tree shows how a number can be split into smaller factors. At each split, the two new numbers multiply to make the number above them. Keep splitting any composite numbers. Stop when every branch ends in a prime. Multiplication facts help you build and check the tree.

What you will learn

Recognize prime and composite numbers used in a factor tree.
Build a factor tree by splitting composite numbers into factors.
Tell when a factor tree is complete.
List and check every prime at the ends of a tree.

1. Review the numbers in a factor tree

A factor is a whole number that can be multiplied by another whole number to make a product. For example,

3

and

4

are factors of

12

because

3 \times 4 = 12

. The product is the answer to a multiplication question.

A prime number is a whole number greater than

1

with exactly two factors:

1

and itself. For example,

5

is prime because its only factors are

1

and

5

. A composite number is a whole number greater than

1

with more than two factors. For example,

12

is composite because

3

and

4

multiply to make it.

The number

1

is neither prime nor composite. A factor tree starts with a composite number. Each split replaces one number with two factors. The numbers at the ends of the branches are called the ends of the tree. A prime factor is a prime number at one of those ends.

The two factors at a split must multiply to make the number being split.
A prime number can end a branch.
A composite number needs to be split further.

2. Build and check a factor tree

Start with the composite number you want to represent. Choose two factors that multiply to make it, and write them as the next branches. Check the split by multiplying the factors.

If a branch ends in a composite number, split it into two factors. If it ends in a prime number, leave it alone. Repeat until every branch ends in a prime.

For example, split

12

into

3

and

4

. This works because

3 \times 4 = 12

. The

3

is prime, but the

4

is composite. Split

4

into

2

and

2

. The ends are now

3

,

2

, and

2

, all prime.

A number can have more than one correct tree. Another way to start with

12

is to split it into

2

and

6

, then split

6

into

2

and

3

. The trees have different shapes, but the ends are still

2

,

2

, and

3

. Each split must be correct, and every end must be prime.

To check a tree, confirm that the two factors at each split multiply to make the number above them. Then check that every end is prime. If an end is composite, the tree is not complete.

Every split must keep the same product.
Split composite branches, but leave prime branches alone.
Different tree shapes can have the same prime ends.

3. Read the ends and practise

When a tree is complete, list every prime number at its ends. Include a number each time it appears. If two branches end in

2

, write both

2

s.

Multiply the listed primes to check the starting number. Their product should match it. Also check that each listed number is prime; a matching product alone does not show that the tree is complete.

Try making a tree for

18

and then one for

28

. For

18

, you could begin with

2

and

9

. For

28

, you could begin with

4

and

7

. In each tree, keep splitting composite branches and leave prime branches alone.

Your tree does not need to match someone else’s tree. It is correct when every split uses factors of the number above it and all the ends are prime.

Include repeated prime ends in your list.
Multiply the final primes to check the starting number.
A complete tree has only prime numbers at its ends.

Worked example

Build a tree for 18

Represent

18

with a prime factor tree.

Prime ends for 18

Each bar shows one prime at the end of the tree. The values are stated factors in this example, not measured data.

Choose a factor pair
Split $18$ into $2$ and $9$ . This is valid because their product is $18$ . The $2$ is prime, but the $9$ is composite, so split only the $9$ .
$18 = 2 \times 9$
Finish the composite branch
Split $9$ into $3$ and $3$ . Both are prime, so all the ends of the tree are now prime.
$9 = 3 \times 3$
Check the ends
The ends are $2$ , $3$ , and $3$ . Their product is the starting number, so the tree is correct.
$2 \times 3 \times 3 = 18$

Answer: One correct tree splits

18

into

2

and

9

, then splits

9

into

3

and

3

. Its prime ends are

2

,

3

, and

3

.

Check: The ends are all prime, and their product is

18

.

Worked example

Split two composite branches

Represent

40

with a prime factor tree.

Prime ends for 40

Each bar shows one prime at the end of the tree. Repeated ends are shown separately. The values are stated factors, not measured data.

Split the starting number
Use $4$ and $10$ as factors of $40$ . Both are composite, so both branches need to be split again.
$40 = 4 \times 10$
Split both branches
Split $4$ into $2$ and $2$ . Split $10$ into $2$ and $5$ . Each pair multiplies to make the number it splits.
$4 = 2 \times 2, \quad 10 = 2 \times 5$
Read and check the ends
The ends are $2$ , $2$ , $2$ , and $5$ . They are all prime, and their product is $40$ .
$2 \times 2 \times 2 \times 5 = 40$

Answer: The prime ends are

2

,

2

,

2

, and

5

.

Check: Each split has the correct product, every end is prime, and the ends multiply to

40

.

Worked example

Choose a different first split

Represent

45

with a prime factor tree. Begin with a pair other than

5

and

9

.

Prime ends for 45

Each bar shows one prime at the end of the tree. Repeated ends are shown separately. The values are stated factors, not measured data.

Choose another factor pair
Use $3$ and $15$ . This is a valid split because their product is $45$ . The $3$ is prime, but the $15$ is composite.
$45 = 3 \times 15$
Split the composite branch
Split $15$ into $3$ and $5$ . Both are prime, so every end of the tree is now prime.
$15 = 3 \times 5$
Check the representation
The ends are $3$ , $3$ , and $5$ . Multiplying them gives the starting number.
$3 \times 3 \times 5 = 45$

Answer: One correct tree starts with

3

and

15

, then splits

15

into

3

and

5

. Its prime ends are

3

,

3

, and

5

.

Check: Both splits are valid, all ends are prime, and their product is

45

.

Common mistakes and how to avoid them

Stopping when a branch ends in a composite number.

Correction: Split that branch again. A completed tree ends only in primes.

Choosing factors that do not multiply to the number above them.

Correction: Check every split by multiplying its two factors.

Leaving out a repeated prime when listing the ends.

Correction: Include each end, even when the same prime appears more than once.

Splitting a prime number further.

Correction: Leave the prime as an end of the tree.

Lesson summary

Split a composite number into factors that multiply to make it.
Keep splitting composite branches until all ends are prime.
List every prime end, including repeated primes.
Multiply the ends to check the starting number.

Check your understanding

Question 1

A tree for

24

has ends

2

,

2

,

2

, and

3

. Which check confirms these are its prime factors?

$2 \times 2 \times 2 \times 3 = 24$ , and all four numbers are prime.
$2 + 2 + 2 + 3 = 24$ , and all four numbers are prime.
The ends must all be different, so the tree is not complete.
One end must be $24$ for the tree to be complete.

Show answer and explanation

2 \times 2 \times 2 \times 3 = 24

, and all four numbers are prime.

The ends are prime, and their product is

24

. A prime may appear more than once.

Question 2

A factor tree for

30

has ends

2

,

3

, and

5

. Which branch, if any, should be split further?

None. All the ends are prime.
Only $2$ , because it is even.
Only $3$ , because it has another factor pair.
All three, because every factor must be split.

Show answer and explanation

None. All the ends are prime.

The ends are all prime, and their product is

30

. The tree is complete.

Question 3

A student splits

28

into

4

and

7

. What should the student do next?

Split $4$ into $2$ and $2$ , and leave $7$ as it is.
Split $7$ into $1$ and $7$ , and leave $4$ as it is.
Stop because $4$ and $7$ are both prime.
Change the first split because $4$ and $7$ do not multiply to $28$ .

Show answer and explanation

Split

4

into

2

and

2

, and leave

7

as it is.

The split is valid because

4 \times 7 = 28

. The

4

is composite and splits into

2

and

2

;

7

is already prime.

Key terms

Factor: A whole number multiplied by another whole number to make a product.
Prime number: A whole number greater than $1$ with exactly two factors: $1$ and itself.
Composite number: A whole number greater than $1$ with more than two factors.
Factor tree: A branching way to split composite numbers into factors.
Prime factor: A prime number at the end of a completed factor tree.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B2.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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B2.6 · Represent composite numbers with prime factor trees

Ontario Grade 6 Mathematics

Split composite numbers into factors until every branch ends in a prime

What you will learn

1. Review the numbers in a factor tree

2. Build and check a factor tree

3. Read the ends and practise

Build a tree for 18

Split two composite branches

Choose a different first split

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 6 Math

About this lesson and its review