Grade 10 Least Common Multiple of Algebraic Expressions: Short Notes

10 ශ්‍රේණිය · 2 වාරය · 11 පරිච්ඡේදය · වීජීය ප්‍රකාශනවල කුඩාම පොදු ගුණාකාරය

91%

of Grade 10 term tests include it (49 of 54)

269

marks in past papers (94 questions)

9 min

full lesson with quick checks

🎯 What you'll learn

Find the LCM of numbers using prime factors and highest powers
Find the LCM of algebraic terms such as 6x²y and 4xy³
Find the LCM of expressions that contain brackets
Factorise expressions first, then find their LCM
Use the LCM as the common denominator of algebraic fractions

When do the buses meet again?

At the Kandy bus stand, the Peradeniya bus leaves every 12 minutes and the Katugastota bus every 18 minutes. Both leave together at 6.00 a.m. When do they next leave together? You need the least common multiple. Today we do the same thing with letters, which you'll need for every algebraic fraction.

Recap: LCM of numbers

Write each number as a product of primes:12 = 2² × 318 = 2 × 3²The LCM takes every prime that appears, at its highest power:LCM = 2² × 3² = 36So the buses meet again 36 minutes later, at 6.36 a.m.The LCM is the smallest number that both numbers divide into exactly.

Letters are factors too

An algebraic term is just a product, so split it the same way:6x²y = 2 × 3 × x × x × y4xy³ = 2 × 2 × x × y × y × yNow treat x and y exactly like the primes 2 and 3. For each number factor and each letter, take the highest power that appears in any term.

Line the factors up

Write matching factors in the same column. The LCM row takes every column once: 2 × 2 × 3 × x × x × y × y × y = 12x²y³.

🧮 LCM of algebraic terms

LCM = LCM of the numbers × each letter at its highest power. Do it in three parts: the numbers, then each letter one by one. Check that every term divides exactly into your answer.

2 × 3 × x² × y , 2² × x × y³ → 2² × 3 × x² × y³

✏️ Three terms

Find the LCM of 4a²b, 6ab and 3b².

1Numbers: LCM of 4, 6 and 3 is 12.
2Letter a: powers a², a, none. Highest is a².
3Letter b: powers b, b, b². Highest is b².
4LCM = 12a²b². Check: 12a²b² ÷ 6ab = 2ab, a whole term.

✅ 12a²b²

⚠️ Don't multiply, don't take the lowest

Examiners keep seeing two slips:1. Multiplying the terms: for 2a²b and 4b², writing 8a²b³. That is a common multiple, but not the least one.2. Missing a power: writing 4a²b instead of 4a²b². Take the highest power of every letter.Write each term as a product of factors first and these slips disappear.

Brackets are single factors

In 2(x + 1) and 3(x + 1)², the bracket (x + 1) behaves like one letter. Its highest power is (x + 1)².LCM = 6(x + 1)²But (x + 1) and (x − 1) are different factors, just like x and y. If both appear, both go into the LCM.

✏️ Terms with a bracket

Find the LCM of 4(x − 2), 6x(x − 2) and 3x².

1Numbers: LCM of 4, 6 and 3 is 12.
2Letter x: highest power is x².
3Bracket (x − 2): highest power is (x − 2).
4LCM = 12x²(x − 2).

✅ 12x²(x − 2)

Factorise first

If an expression is not yet a product, factorise it before anything else:• Common factor: 3x + 9 = 3(x + 3)• Difference of squares: x² − 9 = (x + 3)(x − 3)• Trinomial: x² + 5x + 6 = (x + 2)(x + 3)Then line up the factors and take each one at its highest power.

⚠️ x − 3 is not x and 3

A bracket like (x − 3) is one factor. Don't break it into x and 3 and take the LCM of the pieces.Also take care when factorising: x² − 9 is (x + 3)(x − 3), not (x − 3)². Expand your factors quickly to check before you build the LCM.

Why bother? Common denominators

To add 1/(2x) + 1/(3x), the denominators must match. The LCM of 2x and 3x is 6x:1/(2x) = 3/(6x) and 1/(3x) = 2/(6x)So 1/(2x) + 1/(3x) = 5/(6x).The LCM gives the simplest common denominator, so the working stays short. You'll use this in the next chapter.

📌 Remember

• Break every term into factors: numbers, letters and brackets.• LCM = LCM of the numbers × each factor at its highest power.• Factorise expressions first (common factor, a² − b², trinomials).• A shared factor is written once, not twice.• Check: each given term divides exactly into your LCM.

🧾 Quick revision sheet

What is the LCM of two terms?	The simplest expression that both terms divide into exactly.
Which power of each letter goes into the LCM?	The highest power that appears in any of the terms.
LCM of 6x²y and 4xy³	12x²y³
First step for expressions like x² − 9 or 3x + 9	Factorise them: (x + 3)(x − 3) and 3(x + 3).
LCM of 2(x + 1) and 3(x + 1)²	6(x + 1)²
Why do we need the LCM of algebraic expressions?	It is the simplest common denominator for adding or subtracting algebraic fractions.

Now test yourself: 3 quick checks and 2 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 9-minute lesson free ▶ Practise 5 past-paper questions