Factors of a number
Theory
A factor of a number divides into it exactly, with no remainder. Factors are found in pairs that multiply to give the number, and a common factor is one that two numbers share.
A factor of a number is a whole number that divides into it exactly, leaving no remainder. \(6\) is a factor of \(24\) because \(24 \div 6 = 4\).
Factors come in pairs that multiply to give the number: \(24 = 1\times24 = 2\times12 = 3\times8 = 4\times6\). So the factors of \(24\) are \(1, 2, 3, 4, 6, 8, 12, 24\).
A common factor of two numbers is a factor they both have. Both \(1\) and the number itself are always factors.
Every factor belongs to a pair that multiplies to give the number. Listing the pairs finds all the factors:
| Number | Factor pairs | Factors |
|---|---|---|
| \(12\) | \(1\times12,\ 2\times6,\ 3\times4\) | \(1,2,3,4,6,12\) |
| \(16\) | \(1\times16,\ 2\times8,\ 4\times4\) | \(1,2,4,8,16\) |
| \(24\) | \(1\times24,\ 2\times12,\ 3\times8,\ 4\times6\) | \(1,2,3,4,6,8,12,24\) |
How to find all the factors of a number
- Start at \(1\) and check each whole number in turn: does it divide exactly?
- Pair each factor you find with its partner (the number divided by it).
- Stop when the two numbers in a pair meet, then list every factor in order.
Find the pairs: \(18 = 1\times18 = 2\times9 = 3\times6\).
Factors: \(1, 2, 3, 6, 9, 18\).
\(16 = 1\times16 = 2\times8 = 4\times4\).
Factors: \(1, 2, 4, 8, 16\) — that is \(5\) factors (\(4\) is counted once).
\(20\): \(1, 2, 4, 5, 10, 20\)
\(30\): \(1, 2, 3, 5, 6, 10, 15, 30\)
Shared: \(1, 2, 5, 10\) — that is \(4\).
\(6\times7=42\), and \(3\times14=42\) too.
\(14\) is missing, so \(42\) has \(8\) factors.
Common pitfalls
Frequently asked questions
What is a factor of a number?
A whole number that divides into it exactly, with no remainder. \(6\) is a factor of \(24\) because \(24\div 6 = 4\).
How do you find all the factors of a number?
Look for pairs that multiply to give it. For \(24\): \(1\times24, 2\times12, 3\times8, 4\times6\), so the factors are \(1,2,3,4,6,8,12,24\).
What is a common factor?
A factor shared by two numbers. The common factors of \(20\) and \(30\) are \(1, 2, 5, 10\).
Is 1 a factor of every number?
Yes. Every whole number has \(1\) and itself as factors.
How do you know you have found them all?
Work in pairs from \(1\) until the two numbers in a pair meet. Each pair gives two factors, so all pairs give every factor.