LCM and HCF
Overview
LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number-system questions in CG TET Paper I Mathematics. These concepts test a candidate's understanding of divisibility, factors, and multiples—fundamental skills required to teach primary-level arithmetic.
Mastery here also supports related topics like fractions, ratio-proportion, and simplification. The pedagogy angle requires you to understand how children conceptualize factors and multiples through concrete examples before moving to abstract methods.
Success demands knowing multiple methods (prime factorization, division, listing), recognizing when to apply LCM versus HCF, and avoiding common calculation errors under exam pressure.
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 numbers obtained by multiplying a given number by 1, 2, 3, ... Example: Multiples of 4 are 4, 8, 12, 16, ...
- 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(a, b) × LCM(a, b) = a × b. This formula is a frequent exam shortcut.
- HCF ≤ both numbers ≤ LCM always. HCF divides both numbers; both numbers divide the LCM.
- HCF of co-primes is 1; LCM of co-primes is their product.
Formulas / Key Facts
| Concept | Formula / Fact |
|---|---|
| Product Rule | HCF × LCM = Product of the two numbers |
| Finding LCM | LCM = Product ÷ HCF |
| Finding HCF | HCF = Product ÷ LCM |
| HCF of fractions | HCF of numerators ÷ LCM of denominators |
| LCM of fractions | LCM of numerators ÷ HCF of denominators |
| HCF by division | Divide larger by smaller; continue with remainder until remainder = 0; last divisor is HCF |
| LCM by prime factorization | Take highest power of each prime factor |
| HCF by prime factorization | Take lowest power of common prime factors |
Quick fact: If one number is a factor of the other, HCF = smaller number, LCM = larger number.
Worked Examples
Example 1: Find HCF and LCM of 18 and 24
Prime Factorization Method
18 = 2 × 3 × 3 = 2¹ × 3² 24 = 2 × 2 × 2 × 3 = 2³ × 3¹
HCF = Take lowest powers of common primes = 2¹ × 3¹ = 6 LCM = Take highest powers of all primes = 2³ × 3² = 8 × 9 = 72
Verification: HCF × LCM = 6 × 72 = 432; Product = 18 × 24 = 432 ✓
Example 2: Two bells ring at intervals of 12 minutes and 18 minutes. If they ring together at 9:00 AM, when will they ring together again?
Analysis: "Ring together again" means finding LCM (next common occurrence).
LCM of 12 and 18: 12 = 2² × 3 18 = 2 × 3² LCM = 2² × 3² = 4 × 9 = 36 minutes
Answer: They will ring together again at 9:36 AM.
Example 3: The HCF of two numbers is 8 and their LCM is 96. If one number is 32, find the other.
Using the product rule: HCF × LCM = Product of numbers 8 × 96 = 32 × Other number 768 = 32 × Other number Other number = 768 ÷ 32 = 24
Answer: The other number is 24.
Example 4: Find HCF of 2/3 and 4/5
HCF of fractions = HCF of numerators ÷ LCM of denominators HCF of 2 and 4 = 2 LCM of 3 and 5 = 15
Answer: HCF = 2/15
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Confusing when to use LCM vs HCF | LCM = events happening together, common time, same starting point. HCF = dividing, distributing equally, cutting into pieces. |
| Taking highest power for HCF instead of lowest | HCF uses lowest power of common primes only; LCM uses highest power of all primes. |
| Forgetting to include all prime factors in LCM | LCM must include every prime that appears in any number, not just common ones. |
| Applying product rule to three or more numbers | HCF × LCM = Product works only for two numbers. For three numbers, use factorization method. |
| Swapping numerator/denominator formula for fractions | Remember: HCF of fractions has LCM in denominator; LCM of fractions has HCF in denominator. |
| Calculation errors in long division method | Always verify: HCF must divide both numbers exactly; both numbers must divide LCM exactly. |
Quick Reference
- HCF = largest divisor common to all numbers
- LCM = smallest number divisible by all given numbers
- Product rule (two numbers only): HCF × LCM = a × b
- HCF problem keywords: divide, distribute equally, maximum, largest piece
- LCM problem keywords: together again, same time, minimum, smallest common
- Co-prime numbers: HCF = 1, LCM = product