Year 10 Higher · Factors, powers and surds
Write a number as prime factors and find the HCF and LCM
Learn how to break a number into prime factors for the HCF and LCM, and work exactly with surds by simplifying them, expanding brackets and rationalising denominators.
Definition
Every whole number bigger than 1 breaks into a product of primes in exactly one way, and that breakdown exposes its factors and multiples. A surd is a root that is not a whole number, such as √2: its decimal never ends or repeats, so the only exact way to handle it is to keep the root sign. Both ideas are about exactness. Prime factors describe a number completely, and surds keep an answer exact where a decimal would have to round.
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. Break each number into primes with a factor tree; the HCF multiplies the primes the numbers share, and the LCM multiplies every prime that appears, used the most times it appears in either number.
Remember:
- •Multiply surds under one root, add only like surds, and clear a root from the bottom by multiplying top and bottom by the right partner.
Don’t Forget...
- •Writing √a + √b as √(a + b). Roots do not split over addition: √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5, so the two are different numbers.
- •Simplifying with a square factor that is not the largest, such as √48 = 2√12, and stopping there. 2√12 still hides a square factor, since 12 = 4 × 3; the surd is only fully simplified as 4√3, when no square factor remains under the root.
- •Multiplying only the bottom of a fraction when rationalising. Multiplying just the denominator changes the value of the fraction; top and bottom must be multiplied by the same thing so the fraction is only ever multiplied by 1.
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:
Find the HCF and LCM of 60 and 72.
Write each as prime factors: 60 = 2 × 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3.
The shared primes are 2, 2 and 3, so the HCF is 2 × 2 × 3 = 12.
The LCM uses every prime as often as it appears in either number: 2 × 2 × 2 × 3 × 3 × 5 = 360.
Answer: HCF = 12 and LCM = 360
Why It Works:
√a × √b = √(a × b) because both sides are positive and both square to a × b, and only one positive number can do that. The conjugate clears the root because (a + √b)(a - √b) = a × a - b is a difference of two squares in which the surd terms cancel exactly. A fraction keeps its value throughout, because multiplying top and bottom by the same thing is multiplying by 1.
Another Example:
Write 5/(2 + √3) with a whole number on the bottom.
The bottom is 2 + √3, so multiply top and bottom by its conjugate, 2 - √3.
Expand the bottom: (2 + √3)(2 - √3) = 4 - 2√3 + 2√3 - 3 = 1.
Expand the top: 5 × (2 - √3) = 10 - 5√3.
The fraction is now (10 - 5√3)/1, which is just 10 - 5√3.
Answer: 10 - 5√3
Animated Example:
Free Write a number as prime factors and find the HCF and LCM 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