HCF, LCM & Simplification

HCF and LCM underpin many CDS word problems (bells, tiling, meeting points), and simplification (BODMAS) is tested directly. Both reward a clean method over guesswork.

HCF and LCM

• HCF (greatest common divisor) = the largest number dividing all the given numbers.
• LCM (least common multiple) = the smallest number divisible by all of them.
• Key identity for two numbers: HCF x LCM = product of the two numbers.
• For fractions: HCF = HCF(numerators)/LCM(denominators); LCM = LCM(numerators)/HCF(denominators).

Methods

• Prime-factorisation: HCF = product of common factors at lowest powers; LCM = product of all factors at highest powers.
• Division (Euclidean) method for HCF of large numbers.

Simplification (BODMAS)

• Order: Brackets, Of, Division, Multiplication, Addition, Subtraction. Solve brackets first: ( ), then { }, then [ ].

Typical word problems

• 'Bells toll together again' or 'runners meet at the start' -> use LCM.
• 'Largest tile / greatest measure / maximum students' -> use HCF.

Exam Tricks & Tips

• 🎯 Use HCF x LCM = product of two numbers to find a missing quantity fast.
• 🎯 'Together again / simultaneously' signals LCM; 'greatest size / maximum equal groups' signals HCF.
• 🎯 For fractions, remember HCF uses HCF-over-LCM, and LCM uses LCM-over-HCF.
• 🎯 In simplification, always clear the innermost bracket first and treat 'of' as multiplication with priority.
• 🎯 To find the least number leaving the same remainder r with several divisors, compute LCM then add r.
• ❌ Common mistake: applying operations left to right and ignoring BODMAS order (doing addition before multiplication).

Expected exam pattern

Direct HCF/LCM sums, one bell/tile word problem, and 1-2 BODMAS simplifications; 6-10 quick marks.

Quick recap

Master the HCF x LCM identity, the fraction rules, prime-factorisation, the LCM (together)/HCF (greatest) word-problem cues, and strict BODMAS order.

Cover the answer, recall, then check. 11 cards on HCF, LCM and BODMAS.

Q1. Identity linking HCF and LCM of two numbers?
A1. HCF x LCM = product of the two numbers.

Q2. If two numbers are 12 and 18, HCF = 6; what is their LCM?
A2. (12 x 18)/6 = 36.

Q3. HCF of two fractions?
A3. HCF(numerators) / LCM(denominators).

Q4. LCM of two fractions?
A4. LCM(numerators) / HCF(denominators).

Q5. Which do you use for 'bells toll together again'?
A5. The LCM of their intervals.

Q6. Which do you use for 'largest tile that fits exactly'?
A6. The HCF of the dimensions.

Q7. What does BODMAS stand for?
A7. Brackets, Of, Division, Multiplication, Addition, Subtraction.

Q8. Simplify 12 + 6 / 3 x 2.
A8. 12 + (6/3)x2 = 12 + 4 = 16 (division/multiplication before addition).

Q9. How do you get HCF by prime factorisation?
A9. Multiply the common prime factors taken at their lowest powers.

Q10. How do you get LCM by prime factorisation?
A10. Multiply all prime factors taken at their highest powers.

Q11. Least number divisible by 4, 6 and 8?
A11. LCM(4,6,8) = 24.

HCF (Highest Common Factor) and LCM (Lowest Common Multiple), together with the rules of simplification, are workhorse CDS arithmetic tools. They appear directly and also power problems on time, fractions, and gears/bells.

Core idea / what this tests

This tests how to find the HCF and LCM of numbers, the relationship between them, and how to simplify complex expressions using the correct order of operations. Many word problems are really disguised HCF/LCM questions.

Deep explanation

Beginner — definitions and methods

• HCF is the largest number dividing all given numbers exactly; LCM is the smallest number divisible by all of them.
• Prime factorisation method: HCF = product of the lowest powers of common primes; LCM = product of the highest powers of all primes.
• Division method (for HCF of two numbers): divide, then divide the divisor by the remainder repeatedly; the last non-zero remainder is the HCF.

Intermediate — the key relationship

For any two numbers a and b: HCF × LCM = a × b. So if you know three of the four quantities, you can find the fourth. This is a favourite CDS shortcut. Also: the HCF of fractions = HCF(numerators)/LCM(denominators); the LCM of fractions = LCM(numerators)/HCF(denominators).

Advanced — simplification (BODMAS/VBODMAS)

Complex expressions are simplified in a fixed order — VBODMAS: Vinculum (bar), Brackets, Of, Division, Multiplication, Addition, Subtraction. Work brackets inside-out, handle "of" as multiplication, and remember that division and multiplication have equal priority (left to right), as do addition and subtraction. Typical word problems: the largest tile/rope length that fits exactly → HCF; the time when bells ring together / minimum quantity divisible by all → LCM.

Worked example

The LCM of two numbers is 48 and their HCF is 8. If one number is 16, find the other.

Step 1 — use the relationship: HCF × LCM = product of the numbers, so 8 × 48 = 16 × (other).
Step 2 — compute the left side: 8 × 48 = 384.
Step 3 — divide by the known number: other = 384 ÷ 16 = 24.
Check: HCF(16, 24) = 8 ✓ and LCM(16, 24) = 48 ✓.
Answer: the other number is 24. The HCF × LCM = product rule solved it in one step.

CDS relevance

HCF/LCM questions are common and quick, and simplification underlies almost every numerical problem. Recognising whether a word problem wants HCF (largest exact divisor) or LCM (smallest common multiple) is a high-frequency CDS skill that converts wordy problems into one calculation.

Exam tricks & shortcuts

• HCF × LCM = product of the two numbers — the fastest route when one number is unknown.
• Word-problem trigger: "largest… that divides exactly / greatest measure" = HCF; "smallest… divisible by all / bells ring together" = LCM.
• Fractions: HCF = HCF(num)/LCM(den); LCM = LCM(num)/HCF(den).
• Simplify strictly by VBODMAS, brackets inside-out.

Key formulas

• HCF × LCM = product of the two numbers (for two numbers only).
• HCF = product of lowest powers of common primes; LCM = product of highest powers of all primes.
• HCF of fractions = HCF(numerators)/LCM(denominators); LCM of fractions = LCM(numerators)/HCF(denominators).
• Largest number dividing a, b, c leaving same remainder r: HCF(a−b, b−c, c−a).
• Smallest number leaving remainder r with each divisor: LCM(divisors) + r.
• Co-prime numbers ⇒ HCF = 1, LCM = product.

Usage: use HCF×LCM = product to find the missing quantity when three of four are known.

Worked Example

Problem: Find the HCF and LCM of 24 and 36, and verify the relationship HCF × LCM = product of the two numbers.

Solution:
Step 1 — Prime factorise each number:
24 = 2³ × 3
36 = 2² × 3²

Step 2 — HCF (highest common factor) = product of the lowest powers of common primes:
common primes are 2 and 3; lowest powers are 2² and 3¹.
HCF = 2² × 3 = 4 × 3 = 12.

Step 3 — LCM (least common multiple) = product of the highest powers of all primes present:
highest powers are 2³ and 3².
LCM = 2³ × 3² = 8 × 9 = 72.

Step 4 — Verify HCF × LCM = product of the numbers:
HCF × LCM = 12 × 72 = 864.
24 × 36 = 864.
Both sides equal 864 ✓.

Answer: HCF = 12, LCM = 72, and 12 × 72 = 24 × 36 = 864.