LCM and HCF — Study Notes for Assam TET Paper I

Overview

LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form a foundational arithmetic topic that appears consistently in Assam TET Paper I Mathematics. These concepts test a candidate's understanding of divisibility, factors, and multiples — skills essential for teaching primary-level mathematics.

For the exam, you must know how to find LCM and HCF using multiple methods (listing, prime factorisation, division), understand the relationship between LCM and HCF, and apply these concepts to word problems involving real-life situations like scheduling, distribution, and measurement. Questions typically range from direct calculation to application-based problems involving two or three numbers.

Mastering this topic also strengthens your ability to teach children why these concepts matter in everyday situations — dividing items equally, finding common time intervals, or understanding how numbers relate to each other.

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Key Concepts

**Factors** are numbers that divide a given number exactly without leaving a remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12.

**Multiples** are numbers obtained by multiplying a given number by 1, 2, 3, and so on. Multiples of 4 are 4, 8, 12, 16, 20...

**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** are numbers whose HCF is 1 (for example, 8 and 15).

**The product relationship**: For any two numbers a and b, LCM × HCF = a × b. This is a frequently tested formula.

**HCF is always a factor of LCM** — the HCF divides the LCM exactly.

**HCF of given numbers is always less than or equal to the smallest number; LCM is always greater than or equal to the largest number.**

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Formulas / Key Facts

| Concept | Formula / Fact | |---------|---------------| | Product Rule | LCM(a, b) × HCF(a, b) = a × b | | Finding LCM from HCF | LCM = (a × b) ÷ HCF | | Finding HCF from LCM | HCF = (a × b) ÷ LCM | | HCF by Prime Factorisation | Product of common prime factors with lowest powers | | LCM by Prime Factorisation | Product of all prime factors with highest powers | | HCF of co-primes | Always equals 1 | | LCM of co-primes | Always equals the product of the numbers | | Division Method for HCF | Divide larger by smaller repeatedly until remainder is 0; last divisor is HCF |

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Worked Examples

### Example 1: Find HCF and LCM of 18 and 24 using prime factorisation

**Step 1**: Prime factorise both numbers

18 = 2 × 3 × 3 = 2¹ × 3²
24 = 2 × 2 × 2 × 3 = 2³ × 3¹

**Step 2**: Find HCF (common primes with lowest powers)

Common primes: 2 and 3
HCF = 2¹ × 3¹ = 2 × 3 = **6**

**Step 3**: Find LCM (all primes with highest powers)

LCM = 2³ × 3² = 8 × 9 = **72**

**Verification**: 18 × 24 = 432; HCF × LCM = 6 × 72 = 432 ✓

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### Example 2: The HCF of two numbers is 12 and their LCM is 144. If one number is 48, find the other.

**Using the product rule**: LCM × HCF = Product of two numbers

144 × 12 = 48 × other number

1728 = 48 × other number

Other number = 1728 ÷ 48 = **36**

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### Example 3: Word Problem — Three bells ring at intervals of 6, 9, and 12 minutes. If they ring together at 8:00 AM, when will they ring together again?

**Solution**: Find LCM of 6, 9, and 12

Prime factorisation:

6 = 2 × 3
9 = 3²
12 = 2² × 3

LCM = 2² × 3² = 4 × 9 = **36 minutes**

They will ring together again at **8:36 AM**

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### Example 4: Find the greatest number that divides 45 and 75 leaving remainder 3 in each case.

**Step 1**: Subtract the remainder from both numbers

45 − 3 = 42
75 − 3 = 72

**Step 2**: Find HCF of 42 and 72

42 = 2 × 3 × 7
72 = 2³ × 3²
HCF = 2 × 3 = **6**

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Confusing HCF and LCM — taking highest for HCF and lowest for LCM | Remember: HCF = Highest factor (smaller result), LCM = Lowest multiple (larger result). HCF ≤ smallest number; LCM ≥ largest number. | | Using the product rule for three or more numbers directly | The formula LCM × HCF = a × b works only for two numbers. For three numbers, find LCM and HCF separately using factorisation. | | Forgetting to include all prime factors when finding LCM | LCM needs ALL prime factors from ALL numbers with their highest powers, not just the common ones. | | In word problems, finding LCM when HCF is needed (or vice versa) | Use LCM for "together again" or "common time" problems. Use HCF for "greatest number that divides" or "equal distribution" problems. | | Incomplete prime factorisation leading to wrong HCF/LCM | Always factorise completely until you reach prime numbers only. Double-check by multiplying factors back. |

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Quick Reference

**HCF = common factors, lowest powers; LCM = all factors, highest powers**

**LCM × HCF = Product of two numbers** (use this for missing number problems)

**"Ring together again" → Find LCM**

**"Largest number that divides exactly" → Find HCF**

**"Divides leaving same remainder" → Subtract remainder first, then find HCF**

**HCF of co-primes = 1; LCM of co-primes = their product**

LCM and HCF