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Factors and multiples

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Introduction to Factors and Multiples

Have you ever shared sweets equally among friends or arranged chairs in rows? These everyday activities use the ideas of factors and multiples. Understanding these concepts helps us solve many math problems quickly and is very important for competitive exams.

Factors are numbers that divide another number exactly, without leaving anything behind. For example, 3 is a factor of 12 because 12 divided by 3 is exactly 4.

Multiples are numbers you get when you multiply a number by 1, 2, 3, and so on. For example, multiples of 4 are 4, 8, 12, 16, and so on.

In this chapter, you will learn how to find factors and multiples, understand prime and composite numbers, and apply these ideas to solve real-life problems.

Factors

A factor of a number is a number that divides it exactly without leaving a remainder.

For example, let's find factors of 12:

  • 1 divides 12 exactly (12 / 1 = 12)
  • 2 divides 12 exactly (12 / 2 = 6)
  • 3 divides 12 exactly (12 / 3 = 4)
  • 4 divides 12 exactly (12 / 4 = 3)
  • 6 divides 12 exactly (12 / 6 = 2)
  • 12 divides 12 exactly (12 / 12 = 1)

So, the factors of 12 are 1, 2, 3, 4, 6, and 12.

1 12 2 6 3 4

Diagram: Factor pairs of 12 connected by arrows show how factors come in pairs that multiply to 12.

How to Find Factors of a Number

To find factors of a number:

  1. Start with 1 and the number itself (these are always factors).
  2. Check numbers from 2 up to the square root of the number.
  3. If a number divides the given number exactly (no remainder), it is a factor.
  4. Remember to include both numbers in the factor pair.

Multiples

A multiple of a number is the product of that number and any whole number.

For example, multiples of 3 are:

3 x 1 = 3, 3 x 2 = 6, 3 x 3 = 9, 3 x 4 = 12, 3 x 5 = 15, and so on.

Multiples go on infinitely because you can multiply by any whole number.

Multiples of 3 and 4
Multiples of 3 Multiples of 4
34
68
912
1216
1520
1824
2128
2432
2736
3040

From the table, you can see that 12, 24, and 36 are common multiples of 3 and 4.

Why Are Common Multiples Important?

Common multiples help us find numbers that two or more numbers share when multiplied. The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. This is useful when adding or subtracting fractions or solving problems involving repeated events.

Prime and Composite Numbers

A prime number is a number greater than 1 that has only two factors: 1 and itself.

Examples: 2, 3, 5, 7, 11

A composite number is a number greater than 1 that has more than two factors.

Examples: 4, 6, 8, 9, 12

Note: 1 is neither prime nor composite.

graph TD    N[24]    N --> A[2]    N --> B[12]    B --> C[2]    B --> D[6]    D --> E[2]    D --> F[3]

Flowchart: Prime factorization of 24 breaking it down into prime numbers 2, 2, 2, and 3.

Prime Factorization

Prime factorization means expressing a composite number as a product of its prime factors.

For example, 24 can be written as:

24 = 2 x 2 x 2 x 3

This helps in finding the Highest Common Factor (HCF) and LCM of numbers.

Worked Examples

Example 1: Finding Factors of 18 Easy
Find all factors of 18.

Step 1: Start with 1 and 18 (1 x 18 = 18).

Step 2: Check numbers from 2 to \(\sqrt{18} \approx 4.24\).

2 divides 18 exactly (18 / 2 = 9), so 2 and 9 are factors.

3 divides 18 exactly (18 / 3 = 6), so 3 and 6 are factors.

4 does not divide 18 exactly (18 / 4 = 4.5), so 4 is not a factor.

Answer: Factors of 18 are 1, 2, 3, 6, 9, and 18.

Example 2: Listing Multiples of 5 up to 50 Easy
List all multiples of 5 up to 50.

Step 1: Multiply 5 by integers from 1 to 10.

5 x 1 = 5

5 x 2 = 10

5 x 3 = 15

5 x 4 = 20

5 x 5 = 25

5 x 6 = 30

5 x 7 = 35

5 x 8 = 40

5 x 9 = 45

5 x 10 = 50

Answer: Multiples of 5 up to 50 are 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50.

Example 3: Prime Factorization of 36 Medium
Find the prime factors of 36 using a factor tree.

Step 1: Start with 36 and split into two factors: 6 and 6 (since 6 x 6 = 36).

Step 2: Break down 6 into 2 and 3 (both prime).

So, the prime factors are 2, 2, 3, and 3.

Answer: \(36 = 2 \times 2 \times 3 \times 3\)

36 6 6 2 3 2 3
Example 4: Finding the Least Common Multiple (LCM) of 4 and 6 Medium
Find the LCM of 4 and 6.

Method 1: Listing Multiples

Multiples of 4: 4, 8, 12, 16, 20, 24, ...

Multiples of 6: 6, 12, 18, 24, 30, ...

Common multiples: 12, 24, ...

Smallest common multiple is 12.

Method 2: Using Prime Factorization

Prime factors of 4 = 2 x 2

Prime factors of 6 = 2 x 3

Take all prime factors with highest powers:

2 x 2 x 3 = 12

Answer: LCM of 4 and 6 is 12.

Example 5: Word Problem - Sharing INR 120 Equally Among 4 Friends Easy
Four friends want to share INR 120 equally. How much money will each friend get?

Step 1: Find the factor of 120 that corresponds to 4 friends.

Step 2: Divide 120 by 4.

\(120 \div 4 = 30\)

Answer: Each friend will get INR 30.

Formula Bank

Factors
\[ a \mid b \implies \text{a divides b exactly} \]
where: \(a, b\) are integers
Multiples
\[ M = n \times k \]
where: \(n\) = base number, \(k\) = any integer, \(M\) = multiple
Prime Factorization
\[ N = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_m^{a_m} \]
where: \(N\) = composite number, \(p_i\) = prime factors, \(a_i\) = exponents
LCM (Least Common Multiple)
\[ \mathrm{LCM}(a,b) = \frac{|a \times b|}{\mathrm{HCF}(a,b)} \]
where: \(a,b\) are integers

Tips & Tricks

Tip: Use factor pairs to quickly find all factors of a number.

When to use: When listing factors of medium-sized numbers.

Tip: Memorize multiplication tables up to 12 to identify multiples faster.

When to use: During quick calculations and competitive exams.

Tip: Use prime factorization to find HCF and LCM efficiently.

When to use: When solving problems involving common factors or multiples.

Tip: Check divisibility rules (e.g., for 2, 3, 5) to quickly find factors.

When to use: When testing if a number is a factor without performing full division.

Tip: For word problems, translate the scenario into factors or multiples before solving.

When to use: When dealing with real-life applications involving sharing or grouping.

Common Mistakes to Avoid

❌ Confusing factors with multiples.
✓ Remember factors divide the number exactly; multiples are products of the number and integers.
Why: Both involve division and multiplication, causing conceptual mix-up.
❌ Listing only prime factors instead of all factors.
✓ List all factor pairs, not just prime factors, when asked for factors.
Why: Students often stop at prime factorization without combining factors.
❌ Incorrectly identifying prime numbers (e.g., considering 1 as prime).
✓ Recall that 1 is neither prime nor composite; primes have exactly two distinct factors.
Why: Misunderstanding of prime number definition.
❌ Forgetting to check all possible factor pairs.
✓ Always check factors up to the square root of the number to ensure completeness.
Why: Students stop early, missing larger factor pairs.
❌ Mixing up LCM and HCF in problems.
✓ LCM is the smallest common multiple; HCF is the greatest common factor. Use prime factorization to differentiate.
Why: Both concepts involve commonality but differ in application.
Key Concept

Factors vs Multiples

Factors divide the number exactly; multiples are products of the number and integers.

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