LCM and HCF
Overview
LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number theory questions in HP TET Mathematics.
Mastery requires two things: fluency with the methods (prime factorisation, division method, relationship formula) and the ability to recognise which concept a word problem demands. LCM questions typically involve "when will events coincide?" scenarios, while HCF questions involve "largest possible equal parts" situations.
For HP TET, focus on numbers up to 3–4 digits. Speed matters—practice until you can find LCM and HCF of two numbers in under 30 seconds.
Key Concepts
- Factor: A number that divides another exactly. Factors of 12: 1, 2, 3, 4, 6, 12.
- Multiple: The product of a number and any whole number. Multiples of 4: 4, 8, 12, 16, ...
- HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).
- LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers.
- Co-prime numbers: Two numbers whose HCF is 1 (e.g., 8 and 15). Their LCM equals their product.
- Fundamental relationship: For any two numbers a and b: HCF × LCM = a × b
- HCF of given numbers ≤ smallest number; LCM of given numbers ≥ largest number.
- HCF divides LCM: The HCF of any set of numbers always divides their LCM exactly.
Formulas / Key Facts
| Concept | Formula / Fact |
|---|---|
| Product relationship | HCF(a, b) × LCM(a, b) = a × b |
| Finding LCM when HCF known | LCM = (a × b) ÷ HCF |
| Finding HCF when LCM known | HCF = (a × b) ÷ LCM |
| HCF of fractions | HCF of numerators ÷ LCM of denominators |
| LCM of fractions | LCM of numerators ÷ HCF of denominators |
| Co-prime numbers | HCF = 1, so LCM = a × b |
| HCF by division | Divide larger by smaller; divide divisor by remainder; repeat until remainder = 0 |
| Prime factorisation for HCF | Take lowest power of common primes |
| Prime factorisation for LCM | Take highest power of all primes |
Worked Examples
Example 1: Find HCF and LCM of 36 and 48
Prime Factorisation Method
36 = 2² × 3² 48 = 2⁴ × 3¹
HCF = Take lowest powers of common primes = 2² × 3¹ = 4 × 3 = 12
LCM = Take highest powers of all primes = 2⁴ × 3² = 16 × 9 = 144
Verification: HCF × LCM = 12 × 144 = 1728 = 36 × 48 ✓
Example 2: Division Method for HCF of 272 and 119
Step 1: Divide 272 by 119 272 = 119 × 2 + 34
Step 2: Divide 119 by 34 119 = 34 × 3 + 17
Step 3: Divide 34 by 17 34 = 17 × 2 + 0
Remainder is 0, so HCF = 17
Example 3: Word Problem (LCM type)
Three bells ring at intervals of 6, 9, and 12 minutes. If they ring together at 8:00 AM, when will they ring together again?
Find LCM of 6, 9, 12:
6 = 2 × 3 9 = 3² 12 = 2² × 3
LCM = 2² × 3² = 4 × 9 = 36 minutes
They will ring together at 8:36 AM
Example 4: Word Problem (HCF type)
A rectangular room is 18 m long and 12 m wide. Find the largest square tile that can exactly cover the floor.
We need the largest size that divides both dimensions exactly → find HCF.
HCF(18, 12): 18 = 2 × 3² 12 = 2² × 3 HCF = 2 × 3 = 6 m
Largest tile size = 6 m × 6 m
Example 5: HCF and LCM of Fractions
Find HCF and LCM of 2/3 and 4/5
HCF = HCF(2, 4) ÷ LCM(3, 5) = 2 ÷ 15 = 2/15
LCM = LCM(2, 4) ÷ HCF(3, 5) = 4 ÷ 1 = 4
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Confusing when to use LCM vs HCF | LCM = "when will they meet/coincide?" (common occurrence). HCF = "largest equal division/measure." |
| Taking highest powers for HCF | HCF uses lowest powers of common primes only. LCM uses highest powers of all primes. |
| Forgetting that HCF × LCM = product only works for two numbers | For three or more numbers, this formula does NOT apply directly. Use prime factorisation. |
| Not simplifying before applying fraction formulas | Always reduce fractions to lowest terms first before finding HCF/LCM of fractions. |
| In division method, stopping too early | Continue until remainder becomes exactly zero. The last non-zero divisor is HCF. |
Quick Reference
- HCF → largest equal parts; LCM → first common event
- HCF × LCM = Product (only for two numbers)
- HCF: lowest powers of common primes; LCM: highest powers of all primes
- Division method: keep dividing divisor by remainder until remainder = 0
- Co-primes: HCF = 1, LCM = product
- HCF ≤ smallest number; LCM ≥ largest number