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 |
Worked Examples
Example 1: Prime Factorisation Method for LCM and HCF
Find the HCF and LCM of 36 and 60.
Step 1 – Prime factorise each number:
- 36 = 2² × 3²
- 60 = 2² × 3 × 5
Step 2 – HCF: Take common primes with lowest powers.
- Common primes: 2 and 3
- HCF = 2² × 3¹ = 4 × 3 = 12
Step 3 – LCM: Take all primes with highest powers.
- LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180
Verification: HCF × LCM = 12 × 180 = 2160 = 36 × 60 ✓
Example 2: Division Method for HCF
Find the HCF of 84 and 126 using the division method.
Step 1 – Divide the larger by the smaller:
- 126 ÷ 84 = 1, remainder = 42
Step 2 – Divide the previous divisor by the remainder:
- 84 ÷ 42 = 2, remainder = 0
Step 3 – When remainder is 0, the divisor at that stage is the HCF.
- HCF = 42
Example 3: Word Problem
Two bells ring at intervals of 8 minutes and 12 minutes. If they ring together at 9:00 AM, when will they next ring together?
Solution:
- We need the LCM of 8 and 12.
- 8 = 2³; 12 = 2² × 3
- LCM = 2³ × 3 = 24 minutes
They will ring together again at 9:24 AM.
Example 4: Using the Product Relationship
The HCF of two numbers is 6 and their LCM is 90. If one number is 18, find the other.
Using: HCF × LCM = Product of two numbers
- 6 × 90 = 18 × other number
- 540 = 18 × other number
- Other number = 540 / 18 = 30
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Treating 1 as a prime number | 1 is neither prime nor composite—it has only one factor (itself). Prime numbers must have exactly two distinct factors. |
| Confusing HCF and LCM methods | For HCF, pick the lowest powers of common primes. For LCM, pick the highest powers of all primes. |
| Forgetting to include all prime factors in LCM | Even if a prime appears in only one number, it must be included in the LCM. |
| Applying divisibility test for 4 to the last digit only | Check the last two digits, not just the last digit. For example, 312: check 12, which is divisible by 4. |
| Using product rule for three or more numbers directly | The formula HCF × LCM = a × b works only for two numbers. For three numbers, use prime factorisation or stepwise pairing. |
Quick Reference
- 1 is neither prime nor composite.
- 2 is the smallest and only even prime.
- HCF ≤ each number ≤ LCM — HCF divides both; both divide LCM.
- HCF × LCM = Product (valid for exactly two numbers).
- Co-prime numbers have HCF = 1; their LCM = their product.
- Divisibility by 6 = divisibility by 2 AND 3 (test both).