HCF and LCM
Overview
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) form the backbone of number system problems in TNPSC Group IV. These concepts appear directly as standalone questions and indirectly in problems involving fractions, time and work, pipes and cisterns, and scheduling problems.
For TNPSC Group IV, you must master three things: finding HCF and LCM quickly using multiple methods, understanding the relationship between HCF, LCM and the product of numbers, and applying these concepts to word problems involving bells ringing together, circular tracks, or dividing quantities.
The SSLC-standard scope means questions stay within manageable number ranges, but speed and accuracy matter. A student who can find HCF and LCM mentally for two-digit numbers gains significant time advantage.
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 numbers. It's the first common multiple in the multiplication tables of given numbers.
- Co-prime numbers: Two numbers whose HCF is 1. Example: 8 and 15 are co-prime.
- Fundamental relationship: For any two numbers a and b, HCF × LCM = a × b. This is the most important formula for solving problems quickly.
- HCF of fractions: HCF of numerators ÷ LCM of denominators.
- LCM of fractions: LCM of numerators ÷ HCF of denominators.
- HCF always divides LCM: The LCM of any set of numbers is always divisible by their HCF.
- For co-prime numbers: HCF = 1, and LCM = Product of the numbers.
Formulas / Key Facts
| Formula/Fact | Context |
|---|---|
| HCF × LCM = Product of two numbers | Use when one of HCF/LCM is given, find the other |
| LCM = Product ÷ HCF | Quick calculation when HCF is known |
| HCF = Product ÷ LCM | Quick calculation when LCM is known |
| HCF of fractions = HCF of numerators ÷ LCM of denominators | For fraction-based problems |
| LCM of fractions = LCM of numerators ÷ HCF of denominators | For fraction-based problems |
| LCM is always ≥ largest number | Quick elimination of wrong options |
| HCF is always ≤ smallest number | Quick elimination of wrong options |
| If one number is a factor of another, HCF = smaller, LCM = larger | Shortcut for factor pairs |
Methods to find HCF:
- Prime factorization: Take lowest powers of common prime factors
- Division method: Divide larger by smaller, then divisor by remainder, until remainder is 0
- Successive division: For more than two numbers, find HCF of first two, then HCF of result with third
Methods to find LCM:
- Prime factorization: Take highest powers of all prime factors
- Division method: Divide by prime numbers until all quotients become 1
Worked Examples
Example 1: Basic HCF and LCM Find HCF and LCM of 36 and 48.
Solution: Prime factorization:
- 36 = 2² × 3²
- 48 = 2⁴ × 3¹
HCF = Take lowest powers of common factors = 2² × 3¹ = 4 × 3 = 12 LCM = Take highest powers of all factors = 2⁴ × 3² = 16 × 9 = 144
Verification: HCF × LCM = 12 × 144 = 1728 = 36 × 48 ✓
Example 2: Word Problem — Bells Ringing Together Three bells ring at intervals of 4, 6, and 9 minutes. If they ring together at 10:00 AM, when will they next ring together?
Solution: Time for next simultaneous ring = LCM of 4, 6, 9
Prime factorization:
- 4 = 2²
- 6 = 2 × 3
- 9 = 3²
LCM = 2² × 3² = 4 × 9 = 36 minutes
Next ring together = 10:00 AM + 36 minutes = 10:36 AM
Example 3: Using the HCF-LCM Relationship The HCF of two numbers is 12, and their LCM is 180. If one number is 36, find the other.
Solution: Using HCF × LCM = Product of numbers 12 × 180 = 36 × Other number 2160 = 36 × Other number Other number = 2160 ÷ 36 = 60
Example 4: HCF and LCM of Fractions Find HCF and LCM of 2/3, 4/5, and 6/7.
Solution: Numerators: 2, 4, 6 → HCF = 2, LCM = 12 Denominators: 3, 5, 7 → HCF = 1, LCM = 105
HCF of fractions = HCF of numerators ÷ LCM of denominators = 2/105 = 2/105 LCM of fractions = LCM of numerators ÷ HCF of denominators = 12/1 = 12
Common Mistakes
- Confusing HCF and LCM formulas for fractions → Remember: HCF has "HCF on top, LCM below" while LCM has "LCM on top, HCF below." Use the mnemonic: HCF gives smaller result, LCM gives larger.
- Forgetting to verify using HCF × LCM = Product → Always cross-check your answer. If HCF × LCM ≠ Product of the two numbers, you made an error somewhere.
- Using the product formula for more than two numbers → The formula HCF × LCM = Product works ONLY for two numbers. For three or more numbers, use prime factorization method.
- Selecting LCM smaller than the largest given number → LCM can never be smaller than the largest of the given numbers. Use this to eliminate wrong options quickly.
- Taking highest power for HCF instead of lowest → HCF needs lowest powers of common factors only. LCM needs highest powers of all factors. Students often swap these.
Quick Reference
- HCF × LCM = Product of two numbers (not for three or more numbers)
- HCF ≤ Smallest number; LCM ≥ Largest number
- Bells/meetings/events "together again" = Find LCM of time intervals
- "Largest number that divides exactly" = Find HCF
- "Smallest number divisible by all" = Find LCM
- For co-primes: HCF = 1, LCM = Product