LCM and HCF

Overview

LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form the backbone of number theory at the primary level. These concepts appear directly in OTET Paper I Mathematics and also serve as building blocks for fractions, ratio-proportion, and word problems involving time, work, and distribution.

For OTET, you must master three things: (1) the definitions and distinction between LCM and HCF, (2) multiple methods to calculate them, and (3) their application in real-life word problems. Questions typically test your speed in finding LCM/HCF using prime factorisation or division method, and your ability to identify which concept applies in a given situation.

Understanding the relationship between LCM and HCF is equally important, as questions often involve finding one when the other is given, along with the product of two numbers.

Key Concepts

• **Factor**: A number that divides another number exactly (without remainder). Factors of 12: 1, 2, 3, 4, 6, 12.

• **Multiple**: A number obtained by multiplying a given number by any whole number. Multiples of 4: 4, 8, 12, 16, 20...

• **HCF (Highest Common Factor)**: The greatest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor). HCF is always smaller than or equal to the smallest given number.

• **LCM (Lowest Common Multiple)**: The smallest number that is a multiple of two or more numbers. LCM is always greater than or equal to the largest given number.

• **Co-prime numbers**: Two numbers whose HCF is 1 (e.g., 8 and 15). For co-primes, LCM = product of the numbers.

• **Relationship formula**: For any two numbers a and b: LCM × HCF = a × b. This is a frequently tested concept.

• **When to use HCF**: Problems involving division, distribution into equal groups, cutting/measuring with maximum size.

• **When to use LCM**: Problems involving repetition, cycles, finding when events coincide again.

Formulas / Key Facts

| Concept | Formula/Fact | |---------|--------------| | Fundamental relationship | LCM(a, b) × HCF(a, b) = a × b | | Finding one from other | LCM = (a × b) ÷ HCF | | HCF of co-primes | HCF = 1 | | LCM of co-primes | LCM = Product of the numbers | | HCF of consecutive numbers | Always 1 | | LCM of consecutive numbers | Product of the numbers | | HCF ≤ Smallest number | Always true | | LCM ≥ Largest number | Always true | | HCF divides LCM | Always true |

**Prime Factorisation Method:**

• HCF = Product of common prime factors with lowest powers
• LCM = Product of all prime factors with highest powers

Worked Examples

### Example 1: Find HCF and LCM of 24 and 36

**Prime Factorisation:**

• 24 = 2³ × 3¹
• 36 = 2² × 3²

**HCF** = Common primes with lowest powers = 2² × 3¹ = 4 × 3 = **12**

**LCM** = All primes with highest powers = 2³ × 3² = 8 × 9 = **72**

**Verification:** LCM × HCF = 72 × 12 = 864 = 24 × 36 ✓

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### Example 2: Using Division Method — Find HCF of 56 and 98

**Step 1:** Divide larger by smaller

• 98 ÷ 56 = 1, remainder = 42

**Step 2:** Divide previous divisor by remainder

• 56 ÷ 42 = 1, remainder = 14

**Step 3:** Continue until remainder = 0

• 42 ÷ 14 = 3, remainder = 0

**HCF = 14** (the last divisor)

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### Example 3: Word Problem — LCM Application

*Three bells ring at intervals of 6, 8, and 12 minutes. If they ring together at 9:00 AM, when will they ring together again?*

**Solution:** Find LCM of 6, 8, and 12

Prime factorisation:

• 6 = 2 × 3
• 8 = 2³
• 12 = 2² × 3

LCM = 2³ × 3 = 24 minutes

**Answer:** They will ring together again at **9:24 AM**

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### Example 4: Word Problem — HCF Application

*A shopkeeper has 120 apples and 150 oranges. He wants to pack them in boxes containing equal numbers of fruits (either apples or oranges, not mixed). What is the maximum number of fruits in each box?*

**Solution:** Find HCF of 120 and 150

• 120 = 2³ × 3 × 5
• 150 = 2 × 3 × 5²

HCF = 2 × 3 × 5 = **30 fruits per box**

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### Example 5: Using the Relationship Formula

*The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.*

**Solution:** Using LCM × HCF = Product of numbers

• 180 × 12 = 36 × other number
• 2160 = 36 × other number
• Other number = 2160 ÷ 36 = **60**

Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Confusing LCM and HCF — using LCM for division problems | Remember: HCF for "greatest/maximum/largest equal parts"; LCM for "smallest/when will events repeat" | | Forgetting to include all prime factors in LCM | LCM needs ALL primes from BOTH numbers with highest powers, not just common ones | | Taking highest powers for HCF instead of lowest | HCF = lowest powers of COMMON factors only | | Applying LCM × HCF = a × b for three numbers | This formula works only for TWO numbers; for three numbers, use prime factorisation directly | | Not verifying answer using the relationship | Always cross-check: LCM × HCF should equal the product of the two numbers | | Assuming LCM can be smaller than the larger number | LCM is always ≥ the largest given number; if your answer is smaller, recalculate |

Quick Reference

• **HCF = "What's the biggest piece?"** — division, grouping, cutting problems

• **LCM = "When will it happen together again?"** — cycles, repetition, alignment problems

• **LCM × HCF = Product of two numbers** — the most important relationship

• **Prime factorisation:** HCF uses lowest powers of common factors; LCM uses highest powers of all factors

• **Co-prime shortcut:** If HCF = 1, then LCM = product of the numbers

• **Division method for HCF:** Keep dividing until remainder is zero; last divisor is HCF