LCM and HCF — Study Notes for Bihar TET Paper I
Overview
LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number-system problems in Bihar TET Paper I Mathematics. These concepts test a candidate's understanding of divisibility, factors, and multiples — skills essential for teaching primary-level arithmetic.
Mastery here also supports related topics like fractions, ratio-proportion, and simplification.
For classroom teaching, these concepts help children understand why numbers behave the way they do when grouped, shared, or repeated — making this topic both exam-critical and pedagogically significant.
Key Concepts
- Factors are numbers that divide a given number exactly (without remainder). Example: Factors of 12 are 1, 2, 3, 4, 6, 12.
- Multiples are products obtained by multiplying a number by natural numbers. Example: Multiples of 4 are 4, 8, 12, 16, 20...
- HCF (Highest Common Factor) is the largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).
- LCM (Lowest Common Multiple) is the smallest number that is a multiple of two or more numbers.
- Co-prime numbers have HCF = 1. Example: 8 and 15 are co-prime.
- Fundamental relationship: For any two numbers a and b, HCF × LCM = a × b. This formula is a frequent exam shortcut.
- HCF of fractions = HCF of numerators ÷ LCM of denominators.
- LCM of fractions = LCM of numerators ÷ HCF of denominators.
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, LCM = product of the numbers |
| HCF of consecutive numbers | Always 1 |
| LCM of consecutive numbers | Product of the numbers |
Three methods to find HCF:
- Prime factorisation — Take common prime factors with lowest powers
- Division method — Divide larger by smaller, then divisor by remainder, repeat until remainder is 0
- Listing factors — List all factors and pick the highest common one
Two methods to find LCM:
- Prime factorisation — Take all prime factors with highest powers
- Division method — Divide by primes, continue until all quotients become 1
Worked Examples
Example 1: Find HCF and LCM of 18 and 24
Prime factorisation:
- 18 = 2 × 3²
- 24 = 2³ × 3
HCF = Common primes with lowest powers = 2¹ × 3¹ = 6
LCM = All primes with highest powers = 2³ × 3² = 8 × 9 = 72
Verification: HCF × LCM = 6 × 72 = 432 = 18 × 24 ✓
Example 2: Three bells ring at intervals of 4, 6, and 9 minutes. If they ring together at 9:00 AM, when will they ring together again?
Solution: Find LCM of 4, 6, and 9.
- 4 = 2²
- 6 = 2 × 3
- 9 = 3²
LCM = 2² × 3² = 4 × 9 = 36 minutes
Answer: They will ring together at 9:36 AM.
Example 3: Find HCF of 2/3, 4/5, and 6/7.
Formula: HCF of fractions = HCF of numerators ÷ LCM of denominators
- HCF of 2, 4, 6 = 2
- LCM of 3, 5, 7 = 105
Answer: HCF = 2/105
Example 4: The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.
Using formula: HCF × LCM = Product of numbers
- 12 × 180 = 36 × other number
- 2160 = 36 × other number
- Other number = 2160 ÷ 36 = 60
Answer: 60
Common Mistakes
| Wrong Thinking | Correct Approach |
|---|---|
| Taking highest powers for HCF | HCF uses lowest powers of common primes only |
| Taking lowest powers for LCM | LCM uses highest powers of all primes present |
| Applying product formula to three numbers | HCF × LCM = a × b works only for two numbers |
| Confusing HCF/LCM of fractions | Remember: HCF uses HCF on top, LCM on bottom; LCM is opposite |
| Forgetting to verify answer | Always check: HCF must divide both numbers; LCM must be divisible by both |
| In word problems, using HCF when LCM is needed | "Together again" or "at the same time" → LCM; "Largest piece" or "maximum distribution" → HCF |
Quick Reference
- HCF = common primes, lowest powers; LCM = all primes, highest powers
- HCF × LCM = Product of two numbers (most important shortcut)
- "Bells ring together again" = Find LCM
- "Largest tile for floor" or "Maximum equal distribution" = Find HCF
- HCF of fractions: HCF(num) ÷ LCM(den); LCM of fractions: LCM(num) ÷ HCF(den)
- HCF ≤ both numbers ≤ LCM (always true — use for elimination)
- Co-primes: HCF = 1, so LCM = product