Factors and multiples
If one number divides another exactly, with no remainder, it is a factor of that number.
12 ÷ 3 = 4 exactly, so 3 is a factor of 12.
The factors of 12 are 1, 2, 3, 4, 6 and 12.
Turned the other way round, 12 is a multiple of 3, because 3 × 4 = 12. The multiples of 3 are 3, 6, 9, 12, 15, … and they never end.
Some facts to remember:
1 is a factor of every number.
Every number is a factor and a multiple of itself.
A number has only a limited (finite) list of factors, but an endless list of multiples.
A good way to find all the factors of a number is to find them in pairs. For 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. So the factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Prime and composite numbers
A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.
A composite number has more than two factors. Examples: 4, 6, 8, 9, 10.
1 is neither prime nor composite, because it has only one factor.
2 is the only even prime number. Every other even number has 2 as a factor.
The primes below 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
You can find primes using the Sieve of Eratosthenes. Write the numbers from 2 to 100. Circle 2 and cross out every other multiple of 2. Circle 3 and cross out its other multiples. Keep going with the next number that is not crossed out. The circled numbers are the primes.
Divisibility tests
A number is divisible by | if |
|---|---|
2 | its last digit is 0, 2, 4, 6 or 8 |
3 | the sum of its digits is divisible by 3 |
4 | the number formed by its last two digits is divisible by 4 |
5 | its last digit is 0 or 5 |
6 | it is divisible by both 2 and 3 |
9 | the sum of its digits is divisible by 9 |
10 | its last digit is 0 |
Example. Is 5832 divisible by 9? The digit sum is 5 + 8 + 3 + 2 = 18, and 18 is divisible by 9. So yes.
Prime factorisation
Every composite number can be written as a product of primes in exactly one way (apart from the order). This is its prime factorisation.
Example. 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5.
You can do this with a factor tree, or by dividing repeatedly by the smallest prime that works.
HCF: the highest common factor
The HCF of two numbers is the largest number that is a factor of both.
Example. Find the HCF of 24 and 36.
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
The common prime factors are 2 × 2 × 3, so the HCF is 12.
LCM: the lowest common multiple
The LCM of two numbers is the smallest number that is a multiple of both. Take each prime the greatest number of times it appears in either number.
Example. For 24 and 36, take 2 three times (from 24) and 3 twice (from 36): LCM = 2 × 2 × 2 × 3 × 3 = 72.
For any two numbers, HCF × LCM = product of the numbers. Check: 12 × 72 = 864 = 24 × 36. ✓
When to use which
HCF answers questions about splitting into the largest equal groups. For example: what is the longest ribbon that can cut 24 cm and 36 cm pieces exactly?
LCM answers questions about when things happen together again. For example: two bells ring every 24 and 36 minutes. After how long do they ring together?