Year 9 · Properties of number
Find factors, multiples and primes
Learn how to find factors, multiples and primes, write any number as a product of its prime factors, and use those primes to find the HCF and LCM.
Definition
Every whole number greater than 1 is either prime or can be built by multiplying primes together, and there is only one way to do it. A factor divides a number exactly, a multiple is that number times a whole number, and a prime has no factors except 1 and itself. Writing a number as a product of its prime factors lays out exactly how it is built, and that blueprint is what lets you compare two numbers and find what they share.
Why Do I Have to Learn This?
Multiplying is a fast way of adding the same number again and again. It helps with times tables, sharing fairly and working out costs quickly.
Key Rules:
- 1. A factor divides a number exactly, a multiple is in that number's times table, and a prime has exactly two factors: 1 and itself.
- 2. Find factors in pairs, starting with 1 and the number itself, and stop when the pairs begin to repeat.
Remember:
- •Primes are the building blocks: shared primes make the HCF, and every prime at its highest power makes the LCM.
Don’t Forget...
- •Treating 1 as a prime number. A prime needs exactly two different factors, and 1 has only one, so it is not prime and a factor tree never ends at 1.
- •Swapping HCF and LCM, for example answering 360 when asked for the highest common factor of 24 and 90. The HCF divides both numbers so it can never be bigger than either, while the LCM is a multiple of both so it can never be smaller: a quick size check exposes the swap.
- •Stopping a factor tree at a branch such as 9 or 15 that is not yet prime. 9 = 3 x 3 and 15 = 3 x 5 still split, so the product is not all primes and the index form comes out wrong.
Array
Objects set out in rows and columns, used to show multiplication.
3 rows of 4 dots make an array showing 3 × 4 = 12.
Worked Example:
Write 360 as a product of its prime factors.
Split off the smallest prime repeatedly: 360 = 2 x 180, 180 = 2 x 90, 90 = 2 x 45.
45 is odd, so move to the next prime: 45 = 3 x 15 and 15 = 3 x 5.
Every branch now ends in a prime: 2, 2, 2, 3, 3 and 5.
Write the product in index form, collecting the repeats: 2^3 x 3^2 x 5.
Answer: 360 = 2 x 2 x 2 x 3 x 3 x 5 = 2^3 x 3^2 x 5
Why It Works:
Every whole number greater than 1 factorises into primes in exactly one way, so the prime list is a complete description of the number. Any common factor can only be built from primes both numbers contain, and any common multiple must contain every prime either number holds, which is why the shared primes give the HCF and the higher powers give the LCM.
Another Example:
Find the highest common factor and lowest common multiple of 24 and 90.
Write both in prime factor form: 24 = 2 x 2 x 2 x 3 and 90 = 2 x 3 x 3 x 5.
The HCF takes the shared primes: one 2 and one 3 appear in both lists, so HCF = 2 x 3 = 6.
The LCM takes each prime at its higher power: 2 x 2 x 2 x 3 x 3 x 5 = 360.
Check: HCF x LCM = 6 x 360 = 2160, and 24 x 90 = 2160, so the answers agree.
Answer: HCF = 6 and LCM = 360
Animated Example:
Free Find factors, multiples and primes Worksheet
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Before this: Indices (Year 8) · Next: Factors, powers and surds (Year 10 Foundation)
Let’s Practise the Concept
Step 1 of 3
Put them in order
Tap the numbers from smallest to largest.
In order so far: nothing yet