Year 10 Foundation · Factors, powers and surds
Find factors, multiples and primes
Learn how to multiply and divide decimals without a calculator, break numbers into their prime factors to find the HCF and LCM, and simplify surds.
Definition
Every whole number is built by multiplying primes, the numbers whose only factors are 1 and themselves. Writing a number as its prime factors shows everything about what divides it, including what it shares with another number. The same non-calculator skills cover decimals, where the decimal point simply records a scaling by tens. Surds belong here too: roots such as √2 whose exact value no fraction can write, so the root sign is kept and simplified instead of rounded.
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 sits in its times table, and a prime has exactly two factors, 1 and itself.
Remember:
- •Break numbers into primes: the HCF, the LCM and simplifying surds all start from knowing what multiplies to make what.
Don’t Forget...
- •Stopping a factor tree too early and giving 60 = 4 x 15 as the final answer. 4 and 15 are not prime, so this is not yet a product of prime factors: both must split further, giving 2 x 2 x 3 x 5.
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 60 and 72 as products of their prime factors, then find the HCF and LCM of 60 and 72.
Split 60 with a factor tree: 60 = 2 x 30 = 2 x 2 x 15 = 2 x 2 x 3 x 5.
Split 72 the same way: 72 = 2 x 36 = 2 x 2 x 18 = 2 x 2 x 2 x 3 x 3.
Both lists share 2 x 2 x 3, so the HCF is 12.
For the LCM take each prime at its higher power: 2 x 2 x 2 and 3 x 3 from 72, and the 5 from 60, giving 8 x 9 x 5 = 360.
Answer: 60 = 2 x 2 x 3 x 5, 72 = 2 x 2 x 2 x 3 x 3, HCF = 12 and LCM = 360
Why It Works:
Every whole number has exactly one prime factorisation, so the primes two numbers share are precisely their common factors, which is why multiplying the shared primes gives the highest common factor. Decimal methods work because a division is a fraction: multiplying both numbers by 10 scales top and bottom equally and leaves the answer unchanged. √25 x √2 equals √50 because both sides square to 50: square roots respect multiplication.
Another Example:
Simplify √48, then multiply out √3(4 + √3).
Find the biggest square factor of 48: 48 = 16 x 3, so √48 = √16 x √3 = 4√3.
For the bracket, multiply each term inside by √3: √3 x 4 = 4√3, and √3 x √3 = 3.
Put the two terms together: √3(4 + √3) = 4√3 + 3.
Answer: √48 = 4√3, and √3(4 + √3) = 4√3 + 3
Animated Example:
Free Find factors, multiples and primes Worksheet
Download free worksheets and practise today.
Every download has a fresh set of questions.
Before this: Properties of number (Year 9)
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