Year 6: Factor trees, prime factors and index form
Part of the Year 6 hcf, lcm and prime numbers chapter. Chapter overview · whole chapter on one page
Factor trees, prime factors and index form
A factor tree splits a number into two factors. Keep splitting each branch until every branch ends in a prime number. The primes at the ends are the prime factors. 60 can be split in different ways, but the prime factors are always the same.
When a prime is used more than once, we can use index form to shorten it. 2 × 2 × 3 × 5 = 22 × 3 × 5. The small raised number is the power. It tells us how many times the prime is multiplied. 22 means 2 × 2 = 4.
The prime factors of a number are always the same, however you draw the tree. This is why we can say that each number has one set of prime factors.
Key ideas
- A prime factor is a factor that is a prime number. The prime factors of 12 are 2 and 3, because 12 = 2 × 2 × 3.
- Keep splitting until every branch is prime. 1 is not a prime factor.
- Index form uses a power to show repeats: 2 × 2 × 2 = 23.
- Check your answer by multiplying the prime factors back together.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 60. Write them on two branches.60 = 4 × 15
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).4 = 2 × 2
15 = 3 × 5 - Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 3 × 5 = 60
- Write it in index form. Use a small power for a prime that is used more than once.60 = 22 × 3 × 5
Check. Multiply the primes back together: 2 × 2 × 3 × 5 = 60. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 2 × 3 × 7 = 84
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 2 times
3 is used 1 time
7 is used 1 time - Write it in index form. Use a small power for each prime that is used more than once.22 × 3 × 7
Check. Multiply back: 2 × 2 × 3 × 7 = 84. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 90. Write them on two branches.90 = 2 × 45
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).45 = 3 × 15
15 = 3 × 5 - Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 3 × 3 × 5 = 90
- Write it in index form. Use a small power for a prime that is used more than once.90 = 2 × 32 × 5
Check. Multiply the primes back together: 2 × 3 × 3 × 5 = 90. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Watch out
A factor tree must end in prime numbers only. 6 is not prime, so it must be split again: 6 = 2 × 3. Also, a prime factor is a prime, not just any factor. 12 has factors 2, 3, 4, 6 and 12, but its prime factors are only 2 and 3.Exercise 3D Factor trees and prime factors
Non-calculatorKey words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 48. Write them on two branches.48 = 6 × 8
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).6 = 2 × 3
8 = 2 × 4
4 = 2 × 2 - Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 2 × 2 × 3 = 48
- Write it in index form. Use a small power for a prime that is used more than once.48 = 24 × 3
Check. Multiply the primes back together: 2 × 2 × 2 × 2 × 3 = 48. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 12. Write them on two branches.12 = 2 × 6
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).6 = 2 × 3
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 3 = 12
- Write it in index form. Use a small power for a prime that is used more than once.12 = 22 × 3
Check. Multiply the primes back together: 2 × 2 × 3 = 12. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 20. Write them on two branches.20 = 2 × 10
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).10 = 2 × 5
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 5 = 20
- Write it in index form. Use a small power for a prime that is used more than once.20 = 22 × 5
Check. Multiply the primes back together: 2 × 2 × 5 = 20. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 45. Write them on two branches.45 = 3 × 15
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).15 = 3 × 5
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.3 × 3 × 5 = 45
- Write it in index form. Use a small power for a prime that is used more than once.45 = 32 × 5
Check. Multiply the primes back together: 3 × 3 × 5 = 45. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 50. Write them on two branches.50 = 2 × 25
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).25 = 5 × 5
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 5 × 5 = 50
- Write it in index form. Use a small power for a prime that is used more than once.50 = 2 × 52
Check. Multiply the primes back together: 2 × 5 × 5 = 50. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 66. Write them on two branches.66 = 2 × 33
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).33 = 3 × 11
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 3 × 11 = 66
- Write it in index form. Use a small power for a prime that is used more than once.66 = 2 × 3 × 11
Check. Multiply the primes back together: 2 × 3 × 11 = 66. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 2 × 2 × 5 = 40
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 3 times
5 is used 1 time - Write it in index form. Use a small power for each prime that is used more than once.23 × 5
Check. Multiply back: 2 × 2 × 2 × 5 = 40. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 2 × 2 × 3 × 3 = 72
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 3 times
3 is used 2 times - Write it in index form. Use a small power for each prime that is used more than once.23 × 32
Check. Multiply back: 2 × 2 × 2 × 3 × 3 = 72. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 2 × 5 × 5 = 100
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 2 times
5 is used 2 times - Write it in index form. Use a small power for each prime that is used more than once.22 × 52
Check. Multiply back: 2 × 2 × 5 × 5 = 100. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 3 × 3 × 5 = 90
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 1 time
3 is used 2 times
5 is used 1 time - Write it in index form. Use a small power for each prime that is used more than once.2 × 32 × 5
Check. Multiply back: 2 × 3 × 3 × 5 = 90. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Method. Multiply the prime factors together. Use a running total so you do not lose track.
- Write the product. Each prime is one number to multiply.2 × 3 × 3 × 7
- Multiply two at a time. Keep a running total.2 → 6 → 18 → 126
- Write the number. = 126
Check. Check by factor tree or by division: 126 ÷ 2 = 63. ✓
Watch out. Multiply, do not add. 2 × 3 = 6, not 5.
Method. Multiply the prime factors together. Use a running total so you do not lose track.
- Write the product. Each prime is one number to multiply.2 × 2 × 2 × 5
- Multiply two at a time. Keep a running total.2 → 4 → 8 → 40
- Write the number. = 40
Check. Check by factor tree or by division: 40 ÷ 2 = 20. ✓
Watch out. Multiply, do not add. 2 × 3 = 6, not 5.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Multiply the primes you know, then divide the number by the answer to find the one that is missing.
- Multiply the prime factors you know. These are the numbers already in the product.2 × 2 × 7 = 28
- Divide 84 by that answer. The missing prime is what is left over.84 ÷ 28 = 3
- Check that the missing number is prime. A missing prime factor must be a prime number.3 is prime, because its only factors are 1 and 3.
Check. 2 × 2 × 7 × 3 = 84. ✓
Watch out. Check that the missing number really is prime. If it is not, you have made a mistake.
Method. Multiply the prime factors together. Use a running total so you do not lose track.
- Write the product. Each prime is one number to multiply.2 × 3 × 5 × 7
- Multiply two at a time. Keep a running total.2 → 6 → 30 → 210
- Write the number. = 210
Check. Check by factor tree or by division: 210 ÷ 2 = 105. ✓
Watch out. Multiply, do not add. 2 × 3 = 6, not 5.
Method. Multiply the prime factors together. Use a running total so you do not lose track.
- Write the product. Each prime is one number to multiply.2 × 2 × 3 × 5
- Multiply two at a time. Keep a running total.2 → 4 → 12 → 60
- Write the number. = 60
Check. Check by factor tree or by division: 60 ÷ 2 = 30. ✓
Watch out. Multiply, do not add. 2 × 3 = 6, not 5.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Break each number into prime factors. Keep it only if every prime factor is 2 or 3.
- Check each number from 2 to 49. Its prime factors must all be in the list 2, 3. Write its prime factors.2 = 2 3 = 3 4 = 2 × 2 6 = 2 × 3
8 = 2 × 2 × 2 9 = 3 × 3 12 = 2 × 2 × 3 16 = 2 × 2 × 2 × 2
18 = 2 × 3 × 3 24 = 2 × 2 × 2 × 3 27 = 3 × 3 × 3 32 = 2 × 2 × 2 × 2 × 2
36 = 2 × 2 × 3 × 3 48 = 2 × 2 × 2 × 2 × 3 - Count the numbers that pass. Do not count 1. It has no prime factors.14 numbers
Check. Every number in the list has only 2, 3 as prime factors. ✓
Watch out. A number with a prime factor of 5 or 7 does not count, even if 2 and 3 are also factors.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 12. Write them on two branches.12 = 2 × 6
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).6 = 2 × 3
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 3 = 12
- Write it in index form. Use a small power for a prime that is used more than once.12 = 22 × 3
Check. Multiply the primes back together: 2 × 2 × 3 = 12. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 15. Write them on two branches.15 = 3 × 5
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.3 × 5 = 15
- Write it in index form. Use a small power for a prime that is used more than once.15 = 3 × 5
Check. Multiply the primes back together: 3 × 5 = 15. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Each of 2, 3 and 5 must be a factor, so the number must be a multiple of 2 × 3 × 5 = 30.
- Find the multiples of 30 below 100. Each number must have 2, 3 and 5 as factors.30, 60, 90
- Check the prime factors of each one. No other prime may appear.30 = 2 × 3 × 5
60 = 22 × 3 × 5
90 = 2 × 32 × 5 - List the numbers that pass. These are the answers.30, 60 and 90
Check. Each one is 30 times 1, 2 or 3, and 120 is too big. ✓
Watch out. Use the prime factors to check each number, not just the times table.