HCF and LCM
Overview
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) form the backbone of number system problems in TNPSC Group II/IIA. Mastering HCF and LCM is non-negotiable because the calculations are straightforward once you understand the methods, making these reliable scoring opportunities.
The exam tests your ability to find HCF/LCM quickly using prime factorization or division methods, apply these concepts to word problems involving bells ringing together, circular tracks, cutting ropes/sheets, and recognize the relationship between HCF, LCM, and the product of numbers. Speed matters—you should be able to solve most problems in under 90 seconds.
Key Concepts
- HCF (Highest Common Factor): The largest number that divides two or more numbers exactly without leaving a remainder. Also called GCD (Greatest Common Divisor).
- LCM (Lowest Common Multiple): The smallest number that is exactly divisible by two or more given numbers.
- Fundamental Relationship: For any two numbers a and b: HCF × LCM = a × b. This is the most frequently tested formula.
- Co-prime Numbers: Two numbers with HCF = 1 are co-prime. Their LCM equals their product directly.
- HCF of Fractions: HCF = HCF of numerators ÷ LCM of denominators
- LCM of Fractions: LCM = LCM of numerators ÷ HCF of denominators
- For three or more numbers: HCF × LCM ≠ Product of numbers. The relationship only holds strictly for two numbers.
- Division Method Logic: Repeatedly divide the larger number by smaller; the last non-zero remainder is HCF.
Formulas / Key Facts
| Formula/Fact | Context |
|---|---|
| HCF × LCM = Product of two numbers | Core relationship for two numbers only |
| LCM of co-primes = Product of numbers | When HCF = 1 |
| HCF of (a, b, c) = HCF of (HCF of a,b) with c | Finding HCF of multiple numbers |
| LCM of (a, b, c) = LCM of (LCM of a,b) with c | Finding LCM of multiple numbers |
| HCF of fractions = HCF of numerators / LCM of denominators | For fraction problems |
| LCM of fractions = LCM of numerators / HCF of denominators | For fraction problems |
| If HCF(a,b) = H, then a = Hx, b = Hy where x,y are co-prime | Useful for word problems |
| Numbers divisible by both a and b = Multiples of LCM(a,b) | Divisibility problems |
Worked Examples
Example 1: Basic Calculation Find HCF and LCM of 48 and 180.
Prime factorization:
- 48 = 2⁴ × 3
- 180 = 2² × 3² × 5
HCF = Product of lowest powers of common primes = 2² × 3 = 12 LCM = Product of highest powers of all primes = 2⁴ × 3² × 5 = 720
Verification: 12 × 720 = 8640 = 48 × 180 ✓
Example 2: Word Problem (Bells Ringing) Three bells ring at intervals of 12, 15, and 20 minutes. If they ring together at 9:00 AM, when will they ring together again?
Find LCM of 12, 15, 20:
- 12 = 2² × 3
- 15 = 3 × 5
- 20 = 2² × 5
LCM = 2² × 3 × 5 = 60 minutes
They will ring together after 60 minutes = 10:00 AM
Example 3: Using HCF-LCM Relationship The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.
Using: HCF × LCM = Product of numbers 12 × 180 = 36 × Other number Other number = (12 × 180) / 36 = 2160 / 36 = 60
Example 4: Maximum Size Problem Find the greatest length of rope that can exactly measure 7m 35cm, 8m 40cm, and 9m 45cm.
Convert to cm: 735, 840, 945 Find HCF:
- 735 = 3 × 5 × 7²
- 840 = 2³ × 3 × 5 × 7
- 945 = 3³ × 5 × 7
HCF = 3 × 5 × 7 = 105 cm = 1m 5cm
Example 5: Fraction Problem Find LCM of 2/3, 4/9, and 5/6.
LCM of fractions = LCM of numerators / HCF of denominators
- LCM of 2, 4, 5 = 20
- HCF of 3, 9, 6 = 3
LCM = 20/3
Common Mistakes
- Confusing HCF and LCM applications: Using LCM when HCF is needed. Fix: HCF for "greatest/largest that divides" problems; LCM for "smallest/least that is divisible" or "when will events coincide" problems.
- Applying product formula to three numbers: Writing HCF × LCM = a × b × c. Fix: The formula HCF × LCM = Product works only for exactly two numbers.
- Wrong formula for fractions: Mixing up numerator and denominator rules. Fix: Remember "HCF is smaller, so divide by larger (LCM)."
- Forgetting to convert units: Working with mixed units (meters and centimeters). Fix: Always convert to smallest unit first before finding HCF/LCM.
- Not checking if answer makes sense: HCF must be ≤ smaller number; LCM must be ≥ larger number. If your answer violates this, recalculate.
- Missing co-prime condition: When told HCF of two numbers is H, and you write numbers as Hx and Hy, forgetting that x and y must be co-prime leads to wrong answers.
Quick Reference
- HCF = Pick lowest powers of common prime factors
- LCM = Pick highest powers of all prime factors
- HCF × LCM = a × b (only for two numbers)
- "Greatest length/size that divides exactly" → Find HCF
- "Least time/number divisible by all" → Find LCM
- HCF divides LCM always; LCM is always a multiple of HCF