LCM and HCF
Overview
LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number theory at the primary level. These concepts appear directly in JTET Paper I mathematics questions and also underpin problems on fractions, ratio-proportion, time-and-work, and word problems involving grouping or distribution.
For the exam, you must be able to find LCM and HCF using multiple methods (listing, prime factorisation, division), understand when to apply each concept in word problems, and know the key relationship that connects them. Mastery here also helps in simplifying fractions and solving problems on bells ringing together, circular tracks, and equal distribution.
Key Concepts
- Factor: A number that divides another number exactly (without remainder). Factors of 12 are 1, 2, 3, 4, 6, 12.
- Multiple: A number obtained by multiplying a given number by any whole number. Multiples of 4 are 4, 8, 12, 16, 20...
- HCF (Highest Common Factor): The greatest 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).
- Key relationship: For any two numbers a and b, LCM × HCF = a × b. This is exam-critical.
- When to use HCF: Problems involving division, distribution into equal groups, cutting into equal pieces, finding the largest tile size.
- When to use LCM: Problems involving common time, bells ringing together, circular track meetings, finding common multiples.
Formulas / Key Facts
| Formula / Fact | Context |
|---|---|
| LCM × HCF = Product of two numbers | Works only for two numbers; very useful for finding one when the other is known |
| HCF of co-prime numbers = 1 | Numbers like 9 and 16, or any two consecutive numbers |
| LCM of co-prime numbers = their product | Since HCF = 1, LCM = a × b |
| HCF(a, b) ≤ smaller number | HCF can never exceed the smallest of the given numbers |
| LCM(a, b) ≥ larger number | LCM is at least as large as the biggest number |
| HCF of numbers divides their LCM | Always true; useful for verification |
| If a divides b, then HCF = a and LCM = b | Example: HCF(4, 12) = 4, LCM(4, 12) = 12 |
Methods to find HCF:
- Listing factors method
- Prime factorisation — take common prime factors with lowest powers
- Continued division (Euclid's method)
Methods to find LCM:
- Listing multiples method
- Prime factorisation — take all prime factors with highest powers
- Division method (dividing by primes simultaneously)
Worked Examples
Example 1: Find HCF and LCM of 18 and 24
Using Prime Factorisation:
18 = 2 × 3 × 3 = 2¹ × 3²
24 = 2 × 2 × 2 × 3 = 2³ × 3¹
HCF = Product of common primes with lowest powers = 2¹ × 3¹ = 2 × 3 = 6
LCM = Product of all primes with highest powers = 2³ × 3² = 8 × 9 = 72
Verification: LCM × HCF = 72 × 6 = 432 Product of numbers = 18 × 24 = 432 ✓
Example 2: The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.
Using the formula: LCM × HCF = Product of numbers
180 × 12 = 36 × other number
2160 = 36 × other number
Other number = 2160 ÷ 36 = 60
Example 3: Three bells ring at intervals of 4, 6, and 8 minutes. If they ring together at 9:00 AM, when will they ring together again?
Solution: We need the LCM of 4, 6, and 8.
4 = 2²
6 = 2 × 3
8 = 2³
LCM = 2³ × 3 = 8 × 3 = 24 minutes
They will ring together again at 9:24 AM
Example 4: Find the greatest number that divides 42 and 70 leaving remainders 6 and 10 respectively.
Solution: The required number divides (42 − 6) = 36 and (70 − 10) = 60 exactly.
So we need HCF of 36 and 60.
36 = 2² × 3²
60 = 2² × 3 × 5
HCF = 2² × 3 = 4 × 3 = 12
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Confusing when to use LCM vs HCF — using LCM for division problems | HCF for division/distribution (equal parts, largest measure). LCM for common occurrence (bells, tracks, time intervals). |
| Applying LCM × HCF = Product formula for three or more numbers | This formula works only for two numbers. For three numbers, find LCM and HCF separately using prime factorisation. |
| Taking highest power for HCF and lowest power for LCM | It is the opposite: HCF uses lowest powers of common factors; LCM uses highest powers of all factors. |
| Forgetting to subtract remainders in "greatest number with remainder" problems | Always subtract the given remainder from each number first, then find HCF of the results. |
| Not verifying answers | Always cross-check: HCF must divide both numbers; LCM must be divisible by both numbers; LCM × HCF should equal product (for two numbers). |
Quick Reference
- HCF = common factors, lowest powers → for division, equal distribution, largest measure
- LCM = all factors, highest powers → for common time, simultaneous events, least quantity
- LCM × HCF = a × b (only for two numbers)
- HCF ≤ smaller number; LCM ≥ larger number
- Remainder problems: Subtract remainders first, then find HCF
- Co-primes: HCF = 1, LCM = product