Chapter Overview
Factors and multiples reveal how whole numbers are built. Prime factorisation provides a reliable method for finding the highest common factor and least common multiple.
This chapter belongs to Ganita Prakash, Grade 7. Focus on explaining each step, checking whether an answer is reasonable, and comparing more than one solution method.
Learning Objectives
After studying this chapter, you should be able to:
- explain and apply prime factorisation;
- explain and apply highest common factor;
- explain and apply least common multiple;
- explain and apply relationship;
- communicate the reasoning behind a solution clearly;
- check answers using estimation, substitution, or a second method.
Key Concepts
| Concept | Meaning |
|---|---|
| Prime factorisation | Every integer greater than 1 can be expressed as a product of prime numbers, apart from the order of factors. |
| Highest common factor | The HCF uses each common prime factor with the smallest exponent appearing in the numbers. |
| Least common multiple | The LCM uses every required prime factor with the greatest exponent appearing. |
| Relationship | For two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b. |
Detailed Explanation
Prime factorisation
Every integer greater than 1 can be expressed as a product of prime numbers, apart from the order of factors.
Highest common factor
The HCF uses each common prime factor with the smallest exponent appearing in the numbers.
Least common multiple
The LCM uses every required prime factor with the greatest exponent appearing.
Relationship
For two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b.
Do not memorise a rule without testing it on examples. Ask what each number, operation, line, or symbol represents and whether the result fits the original situation.
Important Rules and Formulae
- 1 is neither prime nor composite.
- Co-prime numbers have HCF 1, but each number need not be prime.
- Use HCF for greatest equal grouping and LCM for earliest common repetition.
Worked Examples
Example 1
Problem: Find HCF and LCM of 24 and 36.
Solution: 24 = 2³×3, 36 = 2²×3²; HCF = 2²×3 = 12, LCM = 2³×3² = 72.
Example 2
Problem: Three bells ring every 6, 8 and 12 minutes. When will they next ring together?
Solution: LCM(6,8,12) = 24 minutes.
Show the mathematical statement, substitution or construction step before writing the final answer. Include units wherever the quantity is measured.
Common Mistakes
- Using the largest exponent for HCF.
- Assuming two odd numbers are always co-prime.
- Choosing HCF for a repeating-time problem.
Quick Revision
- Prime factorisation: Every integer greater than 1 can be expressed as a product of prime numbers, apart from the order of factors.
- Highest common factor: The HCF uses each common prime factor with the smallest exponent appearing in the numbers.
- Least common multiple: The LCM uses every required prime factor with the greatest exponent appearing.
- Relationship: For two positive integers
aandb,HCF(a,b) × LCM(a,b) = a × b.
Practice Questions
- Find the HCF of 45 and 60.
- Find the LCM of 9 and 12.
- Are 14 and 25 co-prime?
- The HCF is 6, LCM is 180 and one number is 30. Find the other.
Answers and Explanations
-
-
- Yes, their HCF is 1.
6×180÷30 = 36.
Self-Check
Explain one rule from this chapter in your own words, create a fresh example, solve it, and verify the answer. If the explanation and verification agree, the concept is understood rather than merely memorised.