Chapter 3 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

M3.1.3 Chapter 3

Multiples and factors

Which numbers share out 36 exactly, with nothing left over?

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

  • List the factors of 36 in pairs
  • Find common multiples of 4 and 6

Joseph has 36 stones. He wants to lay them out in equal rows, with none left over. Which row lengths can he use?

A factor of 36 is a whole number that divides into 36 exactly. 4 is a factor of 36, because 36 ÷ 4 = 9. Two factors that multiply to 36, such as 4 and 9, are a factor pair.

Draw it first

Lay out 36 tiles in equal rows

  1. Rectangles of 36 small tiles, each with a label. 1 × 36 is one row of 36 tiles. 2 × 18 is 2 rows of 18. 3 × 12 is 3 rows of 12. 4 × 9 is 4 rows of 9. 6 × 6 is 6 rows of 6.1 × 36

    1 Make 1 row of 36. This shows 1 × 36 = 36. The pair is 1 and 36.

  2. Rectangles of 36 small tiles, each with a label. 1 × 36 is one row of 36 tiles. 2 × 18 is 2 rows of 18. 3 × 12 is 3 rows of 12. 4 × 9 is 4 rows of 9. 6 × 6 is 6 rows of 6.1 × 362 × 18

    2 Try 2 rows. 36 ÷ 2 = 18, so 2 rows of 18 work. The pair is 2 and 18.

  3. Rectangles of 36 small tiles, each with a label. 1 × 36 is one row of 36 tiles. 2 × 18 is 2 rows of 18. 3 × 12 is 3 rows of 12. 4 × 9 is 4 rows of 9. 6 × 6 is 6 rows of 6.1 × 362 × 183 × 124 × 9

    3 Try 3 rows and 4 rows. 3 rows of 12 and 4 rows of 9 both work.

  4. Rectangles of 36 small tiles, each with a label. 1 × 36 is one row of 36 tiles. 2 × 18 is 2 rows of 18. 3 × 12 is 3 rows of 12. 4 × 9 is 4 rows of 9. 6 × 6 is 6 rows of 6.1 × 362 × 183 × 124 × 96 × 6

    4 Try 6 rows. 6 rows of 6 work. This pair is 6 and 6, so we write 6 only once.

What about 5 rows? 36 ÷ 5 leaves 1 over, so 5 is not a factor. After 6 the pairs repeat in reverse, 9 and 4, then 12 and 3. So you can stop at 6.

Worked example

List the factors of 36.

  1. Try the numbers in turn. 36 ÷ 1 = 36, 36 ÷ 2 = 18, 36 ÷ 3 = 12 and 36 ÷ 4 = 9.
  2. Then 36 ÷ 6 = 6, so 6 pairs with itself. After that the pairs only repeat, so stop at 6.
  3. Write every number from the pairs in order, with 6 once. The factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Answer: 1, 2, 3, 4, 6, 9, 12, 18 and 36

A multiple of 4 is a number in the 4 times table. The first ones are 4, 8, 12, 16, 20, 24 and 28. Multiples go on for ever. A factor is never bigger than the number.

The multiples of 6 are 6, 12, 18, 24, 30 and so on. Some numbers are in both lists. 12 and 24 are in both. We call each one a common multiple of 4 and 6.

Two overlapping circles. The left circle is labelled table of 4, with 4, 8, 16, 20, 28. The right circle is labelled table of 6, with 6, 18, 30. The overlap holds 12 and 24.table of 4table of 648162028122461830
Multiples of 4 and of 6, up to 30. The numbers in the middle are in both times tables.
A table with the head table of 4 and table of 6. The rows are 4 and 6; 8 and 12; 12 and 18; 16 and 24; 20 and 30; 24 and 36.table of 4table of 6468121218162420302436
The first six multiples of 4 and of 6. The numbers 12 and 24 are in both columns, and 12 is the first.

Worked example

Find the first three common multiples of 4 and 6.

  1. Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36.
  2. Multiples of 6: 6, 12, 18, 24, 30, 36.
  3. The numbers in both lists are 12, 24 and 36.

Answer: 12, 24 and 36

See it three ways

With things

Take 12 stones. Lay them out in equal rows. You can make 1 row of 12, 2 rows of 6, or 3 rows of 4.

As a picture

Three rectangles of 12 tiles. The first is labelled 1 × 12, the second 2 × 6 and the third 3 × 4.1 × 122 × 63 × 4

In symbols

1 × 12 = 12, 2 × 6 = 12 and 3 × 4 = 12. The factors of 12 are 1, 2, 3, 4, 6 and 12.

Try it

Find the factor pairs of 24

You need: 24 small stones, beans or buttons.

  1. Lay out all 24 in one row.
  2. Make 2 equal rows. Then try 3 rows, 4 rows, 5 rows, 6 rows and 7 rows.
  3. Each time, see if any stones are left over. Write down the row counts that use every stone.

What you will see: Rows of 2, 3, 4 and 6 work, and so do 1 and 24. Rows of 5 and 7 leave stones over. The factor pairs are 1 and 24, 2 and 12, 3 and 8, 4 and 6.

If you cannot try it, this is what happens.
Four rectangles of 24 tiles. They are labelled 1 × 24, 2 × 12, 3 × 8 and 4 × 6.1 × 242 × 123 × 84 × 6

A wrong idea many people have

The factors of 4 are 4, 8, 12, 16 and so on.

Why it feels right: Factors and multiples are both about times tables, and the two words sound alike.

The test that breaks it: Try 4 ÷ 8. You cannot make groups of 8 from 4 stones. But 4 ÷ 1 = 4, 4 ÷ 2 = 2 and 4 ÷ 4 = 1, with nothing left.

What is true: The factors of 4 are 1, 2 and 4. A factor is never bigger than the number. The list 4, 8, 12 holds multiples of 4, and a multiple is never smaller.

Key words

factor
a whole number that divides into another number exactly.
factor pair
two factors that multiply to make the number, such as 4 and 9 for 36.
multiple
a number in the times table of another number.
common multiple
a number that is a multiple of two or more numbers.

Back to your guess

The answer is B: The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

The numbers that share out 36 exactly are 1, 2, 3, 4, 6, 9, 12, 18 and 36. Do not leave out 1 and 36 themselves. The list 36, 72, 108 shows multiples of 36, not factors.

Check yourself

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

  1. Which list shows all the factors of 20?

    Tap your answer.

    1. Not this one. You left out 1 and 20. Every number has 1 and itself as factors, because 1 × 20 = 20. Go to M3.1.3. Try another.

    2. Not this one. These are multiples of 20, not factors. A factor divides into 20, so it is never bigger than 20. Go to M3.1.3. Try another.

    3. Yes, that is right. The pairs are 1 × 20, 2 × 10 and 4 × 5. Write the numbers in order.

    See the answer

    1, 2, 4, 5, 10 and 20

    The pairs are 1 × 20, 2 × 10 and 4 × 5. Write the numbers in order.

    • If you chose A: You left out 1 and 20. Every number has 1 and itself as factors, because 1 × 20 = 20. Go to M3.1.3.
    • If you chose B: These are multiples of 20, not factors. A factor divides into 20, so it is never bigger than 20. Go to M3.1.3.

  2. Which of these numbers is a factor of 42? The picture shows 42 tiles in one row.

    One row of 42 tiles, with the label 42 tiles.42 tiles
    Tap your answer.

    1. Not this one. 42 is between 36 and 45 in the 9 times table, so it is not in it. 42 ÷ 9 = 4 r 6, so 9 is not a factor. Go to M3.1.3. Try another.

    2. Not this one. 15 is smaller than 42, but 42 ÷ 15 leaves a remainder. A factor must divide exactly. Go to M3.1.3. Try another.

    3. Yes, that is right. 42 ÷ 6 = 7 with nothing left over, so 6 is a factor. The other three leave a remainder.

    4. Not this one. 42 is between 40 and 48 in the 8 times table, so it is not in it. 42 ÷ 8 = 5 r 2, so 8 is not a factor. Go to M3.1.3. Try another.

    See the answer

    6

    42 ÷ 6 = 7 with nothing left over, so 6 is a factor. The other three leave a remainder.

    • If you chose A: 42 is between 36 and 45 in the 9 times table, so it is not in it. 42 ÷ 9 = 4 r 6, so 9 is not a factor. Go to M3.1.3.
    • If you chose B: 15 is smaller than 42, but 42 ÷ 15 leaves a remainder. A factor must divide exactly. Go to M3.1.3.
    • If you chose D: 42 is between 40 and 48 in the 8 times table, so it is not in it. 42 ÷ 8 = 5 r 2, so 8 is not a factor. Go to M3.1.3.

  3. Which list shows the factor pairs of 28?

    A rectangle of 28 tiles, 4 rows of 7, with the label 4 × 7 = 28.4 × 7 = 28
    Tap your answer.

    1. Yes, that is right. 28 ÷ 1 = 28, 28 ÷ 2 = 14 and 28 ÷ 4 = 7. 3, 5 and 6 leave a remainder. After 4 the pairs repeat.

    2. Not this one. You left out the pair 1 × 28. Every number makes a pair with 1. Go to M3.1.3. Try another.

    3. Not this one. 3 × 9 = 27, not 28. Check that each pair multiplies to exactly 28. Go to M3.1.3. Try another.

    See the answer

    1 × 28, 2 × 14 and 4 × 7

    28 ÷ 1 = 28, 28 ÷ 2 = 14 and 28 ÷ 4 = 7. 3, 5 and 6 leave a remainder. After 4 the pairs repeat.

    • If you chose B: You left out the pair 1 × 28. Every number makes a pair with 1. Go to M3.1.3.
    • If you chose C: 3 × 9 = 27, not 28. Check that each pair multiplies to exactly 28. Go to M3.1.3.

  4. Which list shows the first five multiples of 7?

    Tap your answer.

    1. Not this one. You started at 14. The first multiple of 7 is 7 itself, because 1 × 7 = 7. Go to M3.1.3. Try another.

    2. Not this one. You added 10 each time. Multiples of 7 go up by 7 each time. Go to M3.1.3. Try another.

    3. Yes, that is right. Count in 7s: 7, 14, 21, 28, 35. These are the first five numbers in the 7 times table.

    See the answer

    7, 14, 21, 28 and 35

    Count in 7s: 7, 14, 21, 28, 35. These are the first five numbers in the 7 times table.

    • If you chose A: You started at 14. The first multiple of 7 is 7 itself, because 1 × 7 = 7. Go to M3.1.3.
    • If you chose B: You added 10 each time. Multiples of 7 go up by 7 each time. Go to M3.1.3.

  5. Which of these numbers is a common multiple of 3 and 5?

    Tap your answer.

    1. Not this one. 25 is in the 5 times table, but 3 does not divide it: 25 ÷ 3 leaves 1. A common multiple must be in both tables. Go to M3.1.3. Try another.

    2. Yes, that is right. Only 30 is in both tables: 3 × 10 = 30 and 5 × 6 = 30.

    3. Not this one. 20 is in the 5 times table, but 3 does not divide it: 20 ÷ 3 leaves 2. A common multiple must be in both tables. Go to M3.1.3. Try another.

    See the answer

    30

    Only 30 is in both tables: 3 × 10 = 30 and 5 × 6 = 30.

    • If you chose A: 25 is in the 5 times table, but 3 does not divide it: 25 ÷ 3 leaves 1. A common multiple must be in both tables. Go to M3.1.3.
    • If you chose C: 20 is in the 5 times table, but 3 does not divide it: 20 ÷ 3 leaves 2. A common multiple must be in both tables. Go to M3.1.3.

  6. Write the first five multiples of 6.

    See the answer

    6, 12, 18, 24 and 30

    Count in 6s: 6, 12, 18, 24, 30.

  7. How many factors does 24 have?

    See the answer

    8 factors

    The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. That is 8 numbers.

  8. From an earlier chapter

    Work out −6 × −7.

    See the answer

    42

    Two numbers with the same sign give a positive answer. 6 × 7 = 42, so −6 × −7 = 42.

Keep it

Copy this into your exercise book.

Factors divide a number exactly and come in pairs: 36 = 4 × 9. Multiples are the times table: 4, 8, 12. Common multiples are in two lists, like 12 and 24 for 4 and 6.

Draw: Draw four rectangles of squares for the factor pairs of 24. Write 1 × 24, 2 × 12, 3 × 8 and 4 × 6 beside them.

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.