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Year 5 Maths Whole Numbers & Place Value

Factors of a number

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Theory

A factor of a number divides it exactly, with no remainder. This Year 5 maths page shows how to find all the factors of a number by working through factor pairs, how to test a possible factor by division, why square numbers have an odd number of factors, and how to find common factors of two numbers.

A factor of a whole number divides into it exactly, leaving no remainder. \(7\) is a factor of \(28\) because \(28 \div 7 = 4\). \(5\) is not a factor of \(28\) because \(28 \div 5 = 5 \text{ r } 3\) (remainder \(3\)).

Factors come in factor pairs: two numbers that multiply to make the number. \(4 \times 7 = 28\), so \(4\) and \(7\) are a factor pair of \(28\).

Every whole number has \(1\) and itself as factors. The factors of \(28\) are \(1\), \(2\), \(4\), \(7\), \(14\), \(28\).

A common factor of two numbers is a factor of both. The largest common factor (also called the highest common factor) is the biggest of them.

Factor rainbow for 28 The factors of 28 in order, 1, 2, 4, 7, 14, 28, joined by arcs into factor pairs: 1 times 28; 2 times 14; 4 times 7, each making 28. 12471428
A factor rainbow for \(28\): each arc joins a factor pair, \(1 \times 28\), \(2 \times 14\) and \(4 \times 7\).
Factor rainbow for 49 The factors of 49 in order, 1, 7, 49, joined by arcs into factor pairs: 1 times 49; 7 times 7, each making 49. 1749
\(49 = 7 \times 7\): the middle factor pairs with itself, so \(49\) has only three factors.
Array of 32 dots in 4 rows of 8 An array of 32 dots arranged in 4 rows of 8, showing that 4 and 8 are a factor pair of 32.
An array of \(4\) rows of \(8\) shows that \(4 \times 8 = 32\), so \(4\) and \(8\) are factors of \(32\).

Factors are linked to multiplication and division:

\[ a \times b = n \ \Rightarrow\ a \text{ and } b \text{ are factors of } n \]
a × b = n
\[ 32 = 1 \times 32 = 2 \times 16 = 4 \times 8 \]
32=1×32=2×16=4×8
\[ 60 \div 7 = 8 \text{ r } 4 \ \Rightarrow\ 7 \text{ is not a factor} \]
60÷7=8 r 4
When to stop. Test \(1, 2, 3, \ldots\) in order. Once the next number to test is bigger than its partner in the last pair you found, you have found every pair.
Odd number of factors. Factors usually come in pairs, so most numbers have an even number of factors. A square number such as \(49 = 7 \times 7\) has one factor that pairs with itself, so it has an odd number of factors.
Even numbers. Every even number has \(2\) as a factor. Being even does not mean that \(4\), \(6\) or \(8\) is a factor: test them by dividing.

How to list all the factors of a number

  1. Start with \(1 \times\) the number.
  2. Test \(2, 3, 4, \ldots\) in order. If one divides exactly, write it with its partner.
  3. Stop when the factors meet in the middle.
  4. Write the list in order from smallest to largest and count them.

How to find common factors

  1. List all the factors of each number.
  2. Circle the numbers that are in both lists.
  3. The largest circled number is the largest common factor.
Example 1 — Missing factor
Find the missing factor: \(\square \times 9 = 54\).
Solution
\(54 \div 9\)\(=\)\(6\)
54÷9=6

The missing factor is \(6\), because \(6 \times 9 = 54\).

Example 2 — Is it a factor?
(i) Is \(6\) a factor of \(32\)?
(ii) Is \(8\) a factor of \(32\)?
Solution
\(32 \div 6\)\(=\)\(5 \text{ r } 2\)
\(32 \div 8\)\(=\)\(4\)
32÷8=4

(i) No, there is a remainder.
(ii) Yes, \(8 \times 4 = 32\).

Example 3 — All the factors
List all the factors of \(28\). How many are there?
Solution
\(1 \times 28\)\(=\)\(28\)
\(2 \times 14\)\(=\)\(28\)
\(4 \times 7\)\(=\)\(28\)
28=1×28=2×14=4×7

\(3\), \(5\) and \(6\) do not divide \(28\), and the next test, \(7\), is already in a pair. The factors are \(1\), \(2\), \(4\), \(7\), \(14\), \(28\), so there are \(6\).

Example 4 — A square number
How many factors does \(49\) have? Why is the number odd?
Solution
\(1 \times 49\)\(=\)\(49\)
\(7 \times 7\)\(=\)\(49\)
7×7=49

The factors are \(1,\ 7,\ 49\): three factors. \(7\) pairs with itself, so it is only counted once.

Example 5 — Many factors
Find all the factors of \(60\).
Solution
\(1 \times 60\)\(=\)\(60\)
\(2 \times 30\)\(=\)\(60\)
\(3 \times 20\)\(=\)\(60\)
\(4 \times 15\)\(=\)\(60\)
\(5 \times 12\)\(=\)\(60\)
\(6 \times 10\)\(=\)\(60\)
60=6×10

\(7\), \(8\) and \(9\) do not divide \(60\). The factors are \(1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(10\), \(12\), \(15\), \(20\), \(30\), \(60\): \(12\) factors.

Example 6 — Common factors
(i) List the common factors of \(32\) and \(48\).
(ii) What is the largest common factor?
Solution
\(32\)\(:\)\(1,\ 2,\ 4,\ 8,\ 16,\ 32\)
\(48\)\(:\)\(1,\ 2,\ 3,\ 4,\ 6,\ 8,\)
\(\)\(\)\(12,\ 16,\ 24,\ 48\)
common factors 1,2,4,8,16

(i) \(1,\ 2,\ 4,\ 8,\ 16\).
(ii) \(16\).

Example 7 — Arranging in rows
A gardener plants \(32\) trees in equal rows. There must be more than one row and more than one tree in each row.
(i) List the possible numbers of rows.
(ii) What is the largest possible number of rows?
Solution

The factors of \(32\) are \(1,\ 2,\ 4,\ 8,\ 16,\ 32\). One row (\(1\)) and one tree per row (\(32\) rows) are not allowed.

\(\text{rows}\)\(:\)\(2,\ 4,\ 8,\ 16\)
32=16×2

(i) \(2\), \(4\), \(8\) or \(16\) rows.
(ii) \(16\) rows of \(2\) trees.

Example 8 — Find the mistake
Mia lists the factors of \(50\) as \(1,\ 2,\ 5,\ 25,\ 50\). What has she missed?
Solution
\(5 \times 10\)\(=\)\(50\)
5×10=50

She missed \(10\), the partner of \(5\). The factors of \(50\) are \(1\), \(2\), \(5\), \(10\), \(25\), \(50\): \(6\) factors.

Common pitfalls

Forgetting 1 and the number itself. Every number has \(1\) and itself as factors. The factors of \(28\) start with \(1\) and end with \(28\).
Missing a partner. When you find a factor, write its partner straight away. If \(5\) is a factor of \(50\), so is \(50 \div 5 = 10\).
Counting a square factor twice. In \(49 = 7 \times 7\) the \(7\) is one factor, not two.
Even does not mean everything divides. \(32\) is even, but \(6\) is not a factor of it: \(32 \div 6 = 5 \text{ r } 2\).

Frequently asked questions

What is a factor?

A factor is a whole number that divides into another number exactly, with no remainder. \(7\) is a factor of \(28\) because \(28 \div 7 = 4\).

What is a factor pair?

A factor pair is two numbers that multiply to make a given number. \(4\) and \(8\) are a factor pair of \(32\) because \(4 \times 8 = 32\).

How do you find all the factors of a number?

Test 1, 2, 3 and so on in order. Each time one divides exactly, write it with its partner. Stop when the factors meet in the middle.

Why do some numbers have an odd number of factors?

Square numbers have one factor that pairs with itself, such as \(7 \times 7 = 49\). That factor is counted once, so the total is odd.

How can you tell a number is not a factor?

Divide. If there is a remainder, it is not a factor. \(60 \div 7\) is \(8\) remainder \(4\), so \(7\) is not a factor of \(60\).

What is the largest common factor?

It is the biggest number that is a factor of both numbers. The largest common factor of \(32\) and \(48\) is \(16\).

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