LCM and HCF — Study Notes for HSSC CET
Overview
LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form a foundational topic in Quantitative Ability for HSSC CET. Questions from this topic appear regularly, either as direct calculation problems or as applications in word problems involving time, work, and measurement scenarios.
This topic tests your understanding of divisibility, factors, and multiples. Mastering LCM and HCF not only helps you score quick marks on straightforward questions but also speeds up your problem-solving in related topics like fractions, ratio-proportion, and time-and-work. The calculations are mechanical once you know the methods, making this a high-scoring area with practice.
Focus on prime factorization method for accuracy and division method for speed.
Key Concepts
- HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor). Example: HCF of 12 and 18 is 6.
- LCM (Lowest Common Multiple): The smallest number that is exactly divisible by two or more numbers. Example: LCM of 4 and 6 is 12.
- Relationship between LCM and HCF: For any two numbers a and b, LCM × HCF = a × b. This formula is crucial for solving problems where one value is missing.
- Co-prime numbers: Two numbers with HCF = 1. Example: 8 and 15 are co-prime. For co-primes, LCM = product of the numbers.
- LCM is always ≥ the largest number: LCM of any set of numbers is at least as large as the biggest number in that set.
- HCF is always ≤ the smallest number: HCF of any set of numbers cannot exceed the smallest number.
- HCF of fractions: HCF of numerators ÷ LCM of denominators.
- LCM of fractions: LCM of numerators ÷ HCF of denominators.
Formulas / Key Facts
| Formula / Fact | Context |
|---|---|
| LCM × HCF = Product of two numbers | Only valid for exactly two numbers |
| HCF of fractions = HCF of numerators / LCM of denominators | Use when dealing with fractional values |
| LCM of fractions = LCM of numerators / HCF of denominators | Use when dealing with fractional values |
| HCF(a, b) = HCF(b, a mod b) | Euclidean algorithm — repeat until remainder is 0 |
| If HCF(a, b) = 1, then LCM(a, b) = a × b | Co-prime numbers property |
| LCM of co-primes = their product | Quick shortcut for co-prime pairs |
| HCF divides both numbers completely | Any common factor must divide the HCF |
| LCM is divisible by both numbers | Any common multiple must be divisible by LCM |
Worked Examples
Example 1: Find HCF and LCM of 24 and 36
Prime Factorization Method:
- 24 = 2³ × 3¹
- 36 = 2² × 3²
HCF = Take lowest powers of common primes = 2² × 3¹ = 4 × 3 = 12
LCM = Take highest powers of all primes = 2³ × 3² = 8 × 9 = 72
Verification: LCM × HCF = 72 × 12 = 864 = 24 × 36 ✓
Example 2: The HCF of two numbers is 12 and their LCM is 144. If one number is 36, find the other.
Using formula: LCM × HCF = Product of two numbers
144 × 12 = 36 × Other number
1728 = 36 × Other number
Other number = 1728 ÷ 36 = 48
Example 3: Find the largest number that divides 125, 218, and 280 leaving remainders 5, 8, and 10 respectively.
The required number divides:
- 125 − 5 = 120
- 218 − 8 = 210
- 280 − 10 = 270
Find HCF of 120, 210, and 270:
- 120 = 2³ × 3 × 5
- 210 = 2 × 3 × 5 × 7
- 270 = 2 × 3³ × 5
HCF = 2¹ × 3¹ × 5¹ = 30
Example 4: Find LCM of 2/3, 4/9, and 5/6
LCM of fractions = LCM of numerators / HCF of denominators
Numerators: 2, 4, 5 → LCM = 20
Denominators: 3, 9, 6 → HCF = 3
LCM = 20/3 = 20/3
Example 5: Three bells ring at intervals of 6, 9, and 12 minutes. If they ring together at 10:00 AM, when will they ring together again?
Find LCM of 6, 9, 12:
- 6 = 2 × 3
- 9 = 3²
- 12 = 2² × 3
LCM = 2² × 3² = 4 × 9 = 36 minutes
They will ring together at 10:36 AM
Common Mistakes
- Confusing LCM and HCF: Students sometimes calculate LCM when HCF is asked and vice versa. → Remember: "divide" problems need HCF; "meeting again" or "together" problems need LCM.
- Using LCM × HCF formula for more than two numbers: The product formula works only for two numbers. → For three or more numbers, use prime factorization directly.
- Wrong power selection in prime factorization: Taking highest power for HCF or lowest power for LCM. → HCF needs LOWEST powers; LCM needs HIGHEST powers.
- Forgetting to subtract remainders: In "largest number dividing with remainders" problems, students divide original numbers. → Always subtract the respective remainders first, then find HCF.
- Fraction LCM/HCF formula reversal: Mixing up the formula — applying LCM to denominators for HCF of fractions. → For HCF of fractions: HCF of tops / LCM of bottoms. For LCM: exactly opposite.
Quick Reference
- HCF = largest common divisor; LCM = smallest common multiple
- Two numbers: LCM × HCF = Product of the numbers
- HCF: pick LOWEST powers; LCM: pick HIGHEST powers
- "Largest number dividing..." = Find HCF (subtract remainders first)
- "Ring together again / meet again" = Find LCM
- HCF of fractions = HCF(numerators) / LCM(denominators)