Prime, Composite, LCM and GCM

Overview

This topic forms the backbone of number theory at the primary and upper-primary level. Questions on prime numbers, composite numbers, divisibility rules, LCM (Least Common Multiple) and HCF/GCM (Highest Common Factor / Greatest Common Measure) appear consistently in MAHA TET Mathematics. These concepts are not only tested directly but also serve as building blocks for fractions, ratio-proportion, and word problems involving time, work, and distribution.

For TET aspirants, mastery here means two things: rock-solid conceptual clarity (definitions, properties, methods) and speed in calculation. Examiners often design questions that look simple but require careful application of divisibility tests or the relationship between LCM and HCF. A teacher who understands these concepts deeply can also diagnose and remedy student errors effectively—a skill tested in pedagogy-linked questions.

Key Concepts

• **Prime number**: A natural number greater than 1 that has exactly two factors—1 and itself. Examples: 2, 3, 5, 7, 11, 13. Note that 2 is the only even prime number; 1 is neither prime nor composite.

• **Composite number**: A natural number greater than 1 that has more than two factors. Examples: 4, 6, 8, 9, 12. Every composite number can be expressed as a product of primes (Fundamental Theorem of Arithmetic).

• **Co-prime (relatively prime) numbers**: Two numbers whose HCF is 1. Examples: 8 and 15 are co-prime even though neither is prime.

• **Twin primes**: Pairs of primes differing by 2, such as (3, 5), (11, 13), (17, 19).

• **Factors and multiples**: If a divides b exactly, then a is a factor of b and b is a multiple of a.

• **HCF (Highest Common Factor) / GCM (Greatest Common Measure)**: The largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).

• **LCM (Least Common Multiple)**: The smallest positive number that is a multiple of two or more given numbers.

• **Fundamental relationship**: For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.

Formulas / Key Facts

| Concept | Formula / Rule | |---------|----------------| | Product rule | HCF(a, b) × LCM(a, b) = a × b | | Finding LCM from HCF | LCM = (a × b) / HCF | | Finding HCF from LCM | HCF = (a × b) / LCM | | Prime factorisation for HCF | Take the lowest power of each common prime factor | | Prime factorisation for LCM | Take the highest power of each prime factor present | | Divisibility by 2 | Last digit is 0, 2, 4, 6, or 8 | | Divisibility by 3 | Sum of digits divisible by 3 | | Divisibility by 4 | Last two digits form a number divisible by 4 | | Divisibility by 5 | Last digit is 0 or 5 | | Divisibility by 6 | Divisible by both 2 and 3 | | Divisibility by 8 | Last three digits form a number divisible by 8 | | Divisibility by 9 | Sum of digits divisible by 9 | | Divisibility by 11 | Difference of sums of alternate digits is 0 or divisible by 11 |