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 |