What this chapter is about
This chapter teaches you how to find numbers that are shared by two or more numbers. When you look for a common ground between numbers, you search for values that connect them. The main ideas here are common factors and common multiples.
You already know factors and multiples from earlier classes. Now you will learn to find factors and multiples that two or more numbers share. This skill helps when you add or subtract fractions with different denominators. It also helps when you divide things equally among groups or plan events that repeat at different intervals.
After studying this chapter, you should be able to list common factors of given numbers, find the highest common factor (HCF), list common multiples, and find the lowest common multiple (LCM). These tools appear often in daily life and in later mathematics.
Key ideas
- A factor of a number divides it exactly with no remainder. For example, 4 is a factor of 12 because 12 ÷ 4 = 3 exactly.
- A multiple of a number is what you get when you multiply it by 1, 2, 3, and so on. For example, multiples of 5 are 5, 10, 15, 20, …
- Common factors are factors that two or more numbers share. The common factors of 12 and 18 are 1, 2, 3 and 6.
- The Highest Common Factor (HCF) is the largest of all common factors. HCF of 12 and 18 is 6.
- Common multiples are multiples that two or more numbers share. Common multiples of 4 and 6 include 12, 24, 36, …
- The Lowest Common Multiple (LCM) is the smallest common multiple (other than zero). LCM of 4 and 6 is 12.
- Every pair of numbers has 1 as a common factor. If the only common factor is 1, the numbers are called co-prime. For example, 8 and 15 are co-prime.
- LCM is always equal to or greater than the larger number. HCF is always equal to or smaller than the smaller number.
Formulas and facts to remember
- HCF of two numbers ≤ the smaller number.
- LCM of two numbers ≥ the larger number.
- For any two numbers a and b: HCF × LCM = a × b.
- 1 is a common factor of every pair of numbers.
- If one number is a factor of another, then that number itself is the HCF.
- If one number is a multiple of another, then that number itself is the LCM.
Worked examples
Example 1: Find the HCF of 24 and 36.
Step 1: List factors of 24 → 1, 2, 3, 4, 6, 8, 12, 24.
Step 2: List factors of 36 → 1, 2, 3, 4, 6, 9, 12, 18, 36.
Step 3: Common factors → 1, 2, 3, 4, 6, 12.
Step 4: The highest among these is 12.
Answer: HCF of 24 and 36 is 12.
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Example 2: Find the LCM of 6 and 8.
Step 1: Write multiples of 6 → 6, 12, 18, 24, 30, 36, …
Step 2: Write multiples of 8 → 8, 16, 24, 32, 40, …
Step 3: Common multiples → 24, 48, 72, …
Step 4: The smallest common multiple is 24.
Answer: LCM of 6 and 8 is 24.
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Example 3: Priya wants to buy pencils that come in packs of 6 and erasers that come in packs of 8. She wants equal numbers of each. What is the smallest number she can buy?
We need a number that is a multiple of both 6 and 8. That is the LCM.
From Example 2, LCM of 6 and 8 is 24.
Priya should buy 24 pencils (4 packs) and 24 erasers (3 packs).
Common mistakes
- Mixing up HCF and LCM → Remember: HCF is the Highest common Factor (a divider), LCM is the Lowest common Multiple (a product pattern).
- Forgetting that 1 is always a common factor → Include 1 when you list common factors.
- Stopping the multiples list too early and missing the smallest common one → Keep listing until you find a match.
- Thinking LCM can be smaller than both numbers → LCM is always at least as large as the bigger number.
- Confusing factors with multiples → A factor divides the number; a multiple is the number multiplied.
Quick revision
- Factors divide a number exactly; multiples are products of a number with 1, 2, 3, …
- HCF is the greatest number that divides two or more numbers.
- LCM is the smallest number that is a multiple of two or more numbers.
- HCF × LCM = product of the two numbers.
- Numbers with only 1 as their common factor are co-prime.