Chapter Overview

HCF helps when a quantity must be split into the largest equal groups. LCM helps when repeating patterns or events must meet again.

1. Factors and HCF

Factors of 12:

2. HCF by Prime Factorisation

3. Multiples and LCM

Multiples of 4:

Multiples of 6:

Here:

4. LCM by Prime Factorisation

5. HCF-LCM Relationship

For two positive integers:

6. Choosing Between HCF and LCM

A common difficulty is deciding which idea a word problem needs.

Use HCF when you want the largest equal size that divides quantities exactly. Typical clues include largest equal groups, longest possible equal pieces or greatest size of a tile.

Use LCM when you want the first common occurrence of repeating events. Typical clues include together again, next common time or smallest number divisible by several given numbers.

7. Prime Factorisation Method Clearly

Write each number as a product of prime factors.

For HCF, take the prime factors common to all numbers with the smallest powers.

For LCM, take every prime factor needed with the largest powers.

8. Relationship Between HCF and LCM

For two positive whole numbers $a$ and $b$:

Using 24 and 36:

and:

9. HCF in Measurement Problems

Suppose two ribbons are 84 cm and 126 cm long and must be cut into equal pieces of the greatest possible length.

The required length is:

So each piece can be 42 cm long.

10. LCM in Repeating Events

Suppose one light flashes every 6 seconds and another every 8 seconds. If they flash together now, the next time they flash together is after:

seconds.

11. Complete HCF and LCM Notes

Factor Basics

A factor divides a number exactly. For 24, the factors are:

A common factor divides each of two or more numbers. The greatest common factor is called the HCF.

For 18 and 24, common factors are:

so:

Multiple Basics

Multiples are obtained by multiplying a number by whole numbers:

Multiples of 6:

Multiples of 8:

The first positive common multiple is 24, so:

Listing Method

For small numbers, listing factors or multiples is quick. For larger numbers, prime factorisation is more efficient.

Prime Factorisation

Every composite whole number can be expressed as a product of primes.

Therefore:

and:

Co-prime Numbers

Two numbers are co-prime if their HCF is 1. They do not have to be prime themselves.

For example, 8 and 15 are both composite, but:

so they are co-prime.

Useful Checks

The HCF of two numbers cannot be greater than the smaller number. The LCM cannot be smaller than the larger positive number.

If one number divides the other exactly, then:

• HCF is the smaller number;
• LCM is the larger number.

For example, for 8 and 24:

Application Clues

Use HCF for cutting, grouping, arranging into maximum equal groups.

Use LCM for cycles, bells, repeating schedules, common denominators and first common occurrence.

12. More HCF and LCM Examples

Prime Factors as a Common Language

Prime factorisation reveals the building blocks of a number. For HCF, we keep only prime factors shared by all numbers. For LCM, we collect enough prime factors to build every number involved.

For example:

So:

Quick Reasoning Rules

• HCF of consecutive whole numbers is 1.
• LCM of co-prime numbers is their product.
• If $a$ divides $b$, then HCF is $a$ and LCM is $b$.

Common Mistakes

Quick Revision

Practice Questions

Additional Practice

Chapter Summary