Factors, multiples and primes (AQA GCSE Maths): Revision Notes
Factors, multiples and primes
Understanding factors
A factor of a number is any whole number that divides into it with no remainder left over.
Every number has at least two factors: 1 and the number itself. For example, the number 12 has the factors 1, 2, 3, 4, 6, and 12 because each of these numbers divides into 12 exactly.
When finding factors, you're looking for numbers that create whole number results when you divide. If there's any remainder, then it's not a factor!
Factor pairs
Factors work in pairs because they represent a multiplication that gives the original number. Understanding these pairs helps you find factors more systematically.
Worked Example: Finding Factor Pairs for 12
To find all factor pairs for 12:
- 1 × 12 = 12
- 2 × 6 = 12
- 3 × 4 = 12
Notice how each pair multiplies to give the original number 12.
Common factors
A common factor is a number that is a factor of two or more different numbers. For instance, 2 is a common factor of both 6 and 12 because it divides into both numbers exactly.
Finding common factors is useful for simplifying fractions and solving various mathematical problems. The largest common factor is called the Highest Common Factor (HCF).
Understanding multiples
The multiples of a number are all the results you get when you multiply that number by 1, 2, 3, 4, and so on. These are essentially the numbers that appear in that number's times table.
For example, the multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56...
Common multiples
A common multiple is a number that appears in the times tables of two or more different numbers. For example, 12 is a common multiple of both 3 and 4 because 12 appears in both their times tables.
The smallest common multiple of two or more numbers is called the Lowest Common Multiple (LCM). This is particularly useful when working with fractions.
Prime numbers
A prime number has exactly two factors - it can only be divided by 1 and by itself with no remainder.
The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Remember that 1 is not a prime number because it only has one factor (itself). This is a crucial point that many students forget!
The number 2 is special because it's the only even prime number. All other prime numbers are odd.
Factor trees
Factor trees provide a systematic method for finding the prime factors of any number. This visual method helps break down composite numbers into their basic prime building blocks.
Worked Example: Factor Tree for 60
Step 1: Start with 60 and choose any factor pair 60 = 6 × 10
Step 2: Continue breaking down non-prime factors
- 6 = 2 × 3 (both prime - circle them)
- 10 = 2 × 5 (both prime - circle them)
Step 3: Write the final answer using all circled prime numbers 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Step 4: Check your answer 2² × 3 × 5 = 4 × 3 × 5 = 60 ✓
Here's how to create a factor tree systematically:
- Start with your number and choose any factor pair
- Circle any prime factors as you find them
- Continue breaking down non-prime factors until every branch ends with a prime number
- Write the final answer using all the circled prime numbers
Writing in index form
When you have repeated prime factors, you can write them using indices (powers). For example, if 2 appears three times in your factor tree, write it as 2³. This makes your final answer much neater and easier to work with.
Worked examples
Identifying multiples and primes
To identify if a number is a multiple of another, check if it appears in that number's times table. For prime numbers, verify that the number has exactly two factors.
Worked Example: Is 21 a multiple of 7?
Check: Does 21 appear in the 7 times table? 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21
Yes, 21 is a multiple of 7 because 7 × 3 = 21.
Worked Example: Is 17 a prime number?
Step 1: Check if 17 has exactly two factors Step 2: Test divisibility by small primes: 2, 3, 5, 7, 11, 13...
- 17 ÷ 2 = 8.5 (not exact)
- 17 ÷ 3 = 5.67... (not exact)
- 17 ÷ 5 = 3.4 (not exact)
Since 17 is only divisible by 1 and 17, 17 is a prime number.
Using factor trees
When writing a number as a product of its prime factors, use a factor tree to systematically break it down. Check your answer by multiplying all the prime factors back together - you should get your original number.
Common Mistakes to Avoid:
- Forgetting that 1 is not a prime number
- Not checking your final answer by multiplying back
- Stopping the factor tree before all branches end in prime numbers
- Mixing up factors and multiples
Key Points to Remember:
- Factors divide into a number exactly with no remainder
- Multiples are found by multiplying a number by 1, 2, 3, 4... and so on
- Prime numbers have exactly two factors: 1 and themselves
- Factor trees help you find prime factors systematically by breaking numbers down step by step
- Always verify your answers by checking that factors divide exactly or that prime factors multiply back to give the original number