Chapter 6 of 8

All 8 chapters
  1. 1Below zero, readunderstood
  2. 2Calculating with negative numbers, readunderstood
  3. 3Multiples and factors, readunderstood
  4. 4Quick tests for dividing, readunderstood
  5. 5Prime numbers, readunderstood
  6. 6Prime factors, common factors and multiples, readunderstood
  7. 7Squares, cubes and roots, readunderstood
  8. 8Powers of ten, readunderstood

M3.1.6 Chapter 6

Prime factors, common factors and multiples

Can you build every whole number from prime numbers by multiplying?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Write 60 as a product of primes using a factor tree
  • Find the highest common factor of 24 and 36
  • Find the lowest common multiple of 6 and 8

A prime number cannot be split into smaller factors. Every other whole number bigger than 1 can be split. The pieces can be split again, until only primes are left. Those primes are the prime factors. So the answer to the question is yes, for every whole number bigger than 1.

A factor tree shows the splitting. Write the number at the top. Split it into a factor pair. Split each number that is not prime. Stop when every end number is prime.

Draw it first

Build a factor tree for 60

  1. A factor tree for 60. Step 1: 60 is written at the top. Step 2: two branches go from 60 to 6 and 10. Step 3: 6 splits into 2 and 3, and 10 splits into 2 and 5. The end numbers 2, 3, 2 and 5 each have a ring, because they are prime.60

    1 Write 60 at the top.

  2. A factor tree for 60. Step 1: 60 is written at the top. Step 2: two branches go from 60 to 6 and 10. Step 3: 6 splits into 2 and 3, and 10 splits into 2 and 5. The end numbers 2, 3, 2 and 5 each have a ring, because they are prime.60610

    2 Split 60 into a factor pair, such as 6 and 10. Neither is prime, so go on.

  3. A factor tree for 60. Step 1: 60 is written at the top. Step 2: two branches go from 60 to 6 and 10. Step 3: 6 splits into 2 and 3, and 10 splits into 2 and 5. The end numbers 2, 3, 2 and 5 each have a ring, because they are prime.606231025

    3 Split 6 into 2 and 3, and 10 into 2 and 5. Every end number is prime, so stop. They are ringed.

Worked example

Write 60 as a product of primes.

  1. Split 60 into 6 × 10. Check: 6 × 10 = 60.
  2. Split 6 into 2 × 3, and 10 into 2 × 5.
  3. The end numbers are 2, 3, 2 and 5. Write them with × between them, smallest first.

Answer: 60 = 2 × 2 × 3 × 5

A second factor tree for 60. This time 60 splits into 4 and 15. 4 splits into 2 and 2, and 15 splits into 3 and 5.604221535
A different start, 4 × 15. The end numbers are the same primes: 2, 2, 3 and 5.

Any way of splitting a number ends with the same primes. There is only one list of prime factors for each number. That is why we can use it to compare numbers.

The highest common factor (HCF) of two numbers is the biggest number that is a factor of both. Take each prime as many times as both numbers have it, and multiply. If they share no prime, the HCF is 1.

Two factor trees. The top tree is for 24. It splits into 4 and 6, then 4 into 2 and 2, and 6 into 2 and 3. The bottom tree is for 36. It splits into 4 and 9, then 4 into 2 and 2, and 9 into 3 and 3.2442262336422933
24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. Both have 2, 2 and 3.

Worked example

Find the highest common factor of 24 and 36.

  1. The primes of 24 are 2 × 2 × 2 × 3. Check: 2 × 2 × 2 × 3 = 24.
  2. The primes of 36 are 2 × 2 × 3 × 3. Check: 2 × 2 × 3 × 3 = 36.
  3. The shared primes are 2, 2 and 3. 2 × 2 × 3 = 12.

Answer: 12

The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both. List the multiples of each number and find the first one they share.

Two overlapping circles. The left circle is labelled table of 6, with 6, 12, 18. The right circle is labelled table of 8, with 8, 16. The overlap holds 24.table of 6table of 86121824816
The multiples of 6 and of 8, up to 24. Only one number is in both so far.

Worked example

Find the lowest common multiple of 6 and 8.

  1. Multiples of 6: 6, 12, 18, 24, 30.
  2. Multiples of 8: 8, 16, 24, 32.
  3. The first number that is in both lists is 24.
  4. So 24 is in both tables. Check: 24 ÷ 6 = 4 and 24 ÷ 8 = 3.

Answer: 24

You can also use primes. 6 = 2 × 3 and 8 = 2 × 2 × 2. For each prime, take the most times it appears in either number. That gives 2 × 2 × 2 × 3 = 24.

See it three ways

With things

Kofi has two strips of cloth, 24 cm and 36 cm long. He cuts both into equal pieces with none left over. What is the longest piece he can use?

As a picture

Two bars. The top bar is labelled 24 cm: 2 pieces of 12 cm and is cut into 2 equal pieces. The bottom bar is labelled 36 cm: 3 pieces of 12 cm and is cut into 3 equal pieces. The pieces in both bars are the same length.24 cm: 2 pieces of 12 cm36 cm: 3 pieces of 12 cm

In symbols

24 = 2 × 12 and 36 = 3 × 12. The highest common factor of 24 and 36 is 12.

Try it

Make a factor tree for 84

You need: a sheet of paper, a pencil.

  1. Write 84 at the top.
  2. Split it into any factor pair, such as 4 × 21. Split each number that is not prime.
  3. Stop when every end number is prime. Write the primes with × between them.
  4. Start again with a different pair, such as 6 × 14. Do you get the same primes?

What you will see: Both trees end with 2, 2, 3 and 7. 2 × 2 × 3 × 7 = 84.

If you cannot try it, this is what happens.
A factor tree for 84. 84 splits into 4 and 21. 4 splits into 2 and 2, and 21 splits into 3 and 7.844222137

A wrong idea many people have

The lowest common multiple of 6 and 8 is 2, because 2 goes into both.

Why it feels right: The words "lowest" and "common" sound like the smallest number that the two numbers share.

The test that breaks it: Is 2 in the 6 times table? The multiples of 6 are 6, 12, 18, and so on. None is as small as 2.

What is true: A multiple is never smaller than the number. 2 is a common factor of 6 and 8. The lowest common multiple is 24.

Key words

prime factors
the factors of a number that are prime numbers.
factor tree
a diagram that splits a number into smaller factors until only primes are left.
highest common factor
the biggest number that is a factor of two numbers (HCF).
lowest common multiple
the smallest number that is a multiple of two numbers (LCM).

Back to your guess

The answer is B: 60 = 2 × 2 × 3 × 5, and no other list of primes multiplies to 60.

Prime factors must all be prime. 6, 10 and 30 are not prime, so keep splitting. You end with 60 = 2 × 2 × 3 × 5. Whichever way you split, the primes are the same.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which of these is 36 written as a product of primes?

    Tap your answer.

    1. Not this one. These multiply to 36, but 4 and 9 are not prime. Split 4 into 2 × 2 and 9 into 3 × 3. Go to M3.1.6. Try another.

    2. Not this one. 18 is not prime. Split it: 18 = 2 × 9, and 9 = 3 × 3. Keep going until every number is prime. Go to M3.1.6. Try another.

    3. Yes, that is right. 2 × 2 × 3 × 3 = 36, and 2 and 3 are both prime. The other two lists have numbers that can still be split.

    See the answer

    2 × 2 × 3 × 3

    2 × 2 × 3 × 3 = 36, and 2 and 3 are both prime. The other two lists have numbers that can still be split.

    • If you chose A: These multiply to 36, but 4 and 9 are not prime. Split 4 into 2 × 2 and 9 into 3 × 3. Go to M3.1.6.
    • If you chose B: 18 is not prime. Split it: 18 = 2 × 9, and 9 = 3 × 3. Keep going until every number is prime. Go to M3.1.6.

  2. Which of these is 90 written as a product of primes?

    Tap your answer.

    1. Not this one. 9 is not prime. Split it into 3 × 3. Every number in the list must be prime. Go to M3.1.6. Try another.

    2. Yes, that is right. 90 = 9 × 10. Then 9 = 3 × 3 and 10 = 2 × 5. In order, the primes are 2, 3, 3 and 5.

    3. Not this one. 9 and 10 multiply to 90, but neither is prime. Split 9 into 3 × 3 and 10 into 2 × 5. Go to M3.1.6. Try another.

    See the answer

    2 × 3 × 3 × 5

    90 = 9 × 10. Then 9 = 3 × 3 and 10 = 2 × 5. In order, the primes are 2, 3, 3 and 5.

    • If you chose A: 9 is not prime. Split it into 3 × 3. Every number in the list must be prime. Go to M3.1.6.
    • If you chose C: 9 and 10 multiply to 90, but neither is prime. Split 9 into 3 × 3 and 10 into 2 × 5. Go to M3.1.6.

  3. Look at the factor tree. What number is hidden by the question mark?

    A factor tree for 72. 72 splits into 8 and a number shown as a question mark. 8 splits into 2 and 4, and 4 splits into 2 and 2. The question mark splits into 3 and 3.7282422?33
    Tap your answer.

    1. Yes, that is right. The branches from 72 must multiply back to 72. 8 × ? = 72, so ? = 72 ÷ 8 = 9. It splits into 3 × 3, which is 9.

    2. Not this one. You took 8 away from 72. The two branches multiply to give 72, so 8 × ? = 72. Go to M3.1.6. Try another.

    3. Not this one. You added 8 to 72. The branches of a factor tree multiply back to the number above, they do not add. Go to M3.1.6. Try another.

    See the answer

    9

    The branches from 72 must multiply back to 72. 8 × ? = 72, so ? = 72 ÷ 8 = 9. It splits into 3 × 3, which is 9.

    • If you chose B: You took 8 away from 72. The two branches multiply to give 72, so 8 × ? = 72. Go to M3.1.6.
    • If you chose C: You added 8 to 72. The branches of a factor tree multiply back to the number above, they do not add. Go to M3.1.6.

  4. Find the highest common factor of 18 and 30.

    Tap your answer.

    1. Not this one. 3 is a common factor of 18 and 30, but it is not the highest. 6 is bigger and divides both. Go to M3.1.6. Try another.

    2. Not this one. 90 is the lowest common multiple of 18 and 30. A factor of 18 can never be bigger than 18. Go to M3.1.6. Try another.

    3. Yes, that is right. 18 = 2 × 3 × 3 and 30 = 2 × 3 × 5. They share 2 and 3. 2 × 3 = 6.

    See the answer

    6

    18 = 2 × 3 × 3 and 30 = 2 × 3 × 5. They share 2 and 3. 2 × 3 = 6.

    • If you chose A: 3 is a common factor of 18 and 30, but it is not the highest. 6 is bigger and divides both. Go to M3.1.6.
    • If you chose B: 90 is the lowest common multiple of 18 and 30. A factor of 18 can never be bigger than 18. Go to M3.1.6.

  5. Find the lowest common multiple of 4 and 10.

    Tap your answer.

    1. Not this one. 40 is a common multiple of 4 and 10, but it is not the lowest. 20 is in both tables and is smaller. Go to M3.1.6. Try another.

    2. Yes, that is right. Multiples of 10 are 10, 20, 30. Multiples of 4 are 4, 8, 12, 16, 20. The first one in both lists is 20.

    3. Not this one. 2 is the highest common factor of 4 and 10. A multiple is never smaller than the number itself. Go to M3.1.6. Try another.

    See the answer

    20

    Multiples of 10 are 10, 20, 30. Multiples of 4 are 4, 8, 12, 16, 20. The first one in both lists is 20.

    • If you chose A: 40 is a common multiple of 4 and 10, but it is not the lowest. 20 is in both tables and is smaller. Go to M3.1.6.
    • If you chose C: 2 is the highest common factor of 4 and 10. A multiple is never smaller than the number itself. Go to M3.1.6.

  6. Find the highest common factor of 16 and 40.

    See the answer

    8

    16 = 2 × 2 × 2 × 2 and 40 = 2 × 2 × 2 × 5. They share 2, 2 and 2. 2 × 2 × 2 = 8.

  7. Find the lowest common multiple of 12 and 18.

    See the answer

    36

    Multiples of 12: 12, 24, 36. Multiples of 18: 18, 36. The first one in both lists is 36.

  8. From an earlier chapter

    Is 87 prime? Say why.

    See the answer

    No, 87 = 3 × 29.

    The digit sum is 8 + 7 = 15, which is in the 3 times table. So 3 divides 87, and 87 = 3 × 29.

Keep it

Copy this into your exercise book.

Split a number into a factor pair, and keep splitting until every end is prime. 60 = 2 × 2 × 3 × 5. The HCF is the biggest shared factor. The LCM is the smallest shared multiple.

Draw: Draw a factor tree for 60. Put 6 and 10 below it, then 2, 3, 2 and 5. Ring each prime.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (805 KB). It opens on any phone.