Factors and Multiples

If 36 sweets can be arranged as 4 rows of 9, both 4 and 9 fit exactly into 36. That one arrangement gives a factor pair, and the same fact says 36 is a multiple of each factor.

Parents: let your child predict the pair or shared number before opening the factor lab, then ask what the zero-remainder check proves.

By the end you will be able to explain factors and multiples, test one-digit candidates, list common factors and match common multiples.

Factor pairs show the relationship

A factor divides a number with no remainder. If 4 × 9 = 36, then 4 and 9 are factors of 36, and 36 is a multiple of both 4 and 9. The multiplication statement points in both directions.

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Factor pairs and shared multiples

🤔 Predict first: How many positive factor pairs does 36 have?

3 pairs5 pairs7 pairs

One multiplication fact, two names

$$4\times9=36$$

$$\text{4 and 9 are factors of 36.}$$

$$\text{36 is a multiple of 4 and a multiple of 9.}$$

Test a one-digit factor

For a candidate factor, divide the given number by the candidate and inspect the remainder. For example, 7 is not a factor of 48 because 48 ÷ 7 leaves a remainder. A divisor must be positive; 0 cannot be used as a divisor.

Show the worked solution

Zero-remainder factor check

$$42\div6=7\text{ remainder }0$$

$$6\text{ is a factor of }42$$

Zero-remainder factor check

$$48\div7=6\text{ remainder }6$$

$$7\text{ is not a factor of }48$$

List common factors

List each factor set completely, then keep the entries that appear in both. For 24 and 36, the shared values are 1, 2, 3, 4, 6 and 12.

Common factors of 24 and 36

$$\text{Factors}(24)=\{1,2,3,4,6,8,12,24\}$$

$$\text{Factors}(36)=\{1,2,3,4,6,9,12,18,36\}$$

$$\text{Common factors}=\{1,2,3,4,6,12\}$$

Test a multiple

To test whether a number is a multiple of a one-digit number, check whether it appears in that times table or has a zero remainder when divided. For example, 54 is a multiple of 9 because 54 ÷ 9 = 6.

Show the worked solution

Multiple check

$$48\div8=6\text{ remainder }0$$

$$48=8\times6$$

$$\text{Therefore 48 is a multiple of 8.}$$

Match common multiples

Write positive multiples of each one-digit input and match the lists. Zero is formally a multiple of every non-zero whole number, but when a question asks for the first or smallest common multiple, list the first positive match. For 4 and 6, that match is 12.

Common multiples of 4 and 6

$$\text{Multiples of 4: }4,8,12,16,20,24,28,32$$

$$\text{Multiples of 6: }6,12,18,24,30,36,42,48$$

$$\text{First positive common multiple}=12$$

Practise with worked steps

Practice 1. List every positive factor of 36 in pairs.

Show the factor-pair solution

Factor pairs of 36

$$1\times36=36$$

$$2\times18=36$$

$$3\times12=36$$

$$4\times9=36$$

$$6\times6=36$$

$$\text{Factors: }1,2,3,4,6,9,12,18,36$$

Practice 2. Find every common factor of 24 and 36.

Show the complete-list solution

Keep entries in both lists

$$\text{Factors}(24)=\{1,2,3,4,6,8,12,24\}$$

$$\text{Factors}(36)=\{1,2,3,4,6,9,12,18,36\}$$

$$\text{Common factors}=\{1,2,3,4,6,12\}$$

Practice 3. Find the first positive common multiple of 4 and 6.

Show the matching-multiples solution

Match positive times-table values

$$\text{Multiples of 4: }4,8,12,16,20,24$$

$$\text{Multiples of 6: }6,12,18,24$$

$$\text{First positive match}=12$$

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. If 5 × 7 = 35, which statement is true?

   5 is a multiple of 3535 is a factor of 535 is a multiple of 5

2. Which candidate is a factor of 63?

   478

3. What are all the common factors of 24 and 36?

   1, 2, 3, 4, 6, 121, 2, 3, 62, 4, 8, 12

4. Which number is a multiple of 9?

   445456

5. What is the first positive common multiple of 4 and 6?

   01224

6. Which action is valid when testing whether 0 is a factor of 36?

   Divide 36 by 0Do not use 0 as a divisorCall 0 a factor because it is smaller

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