# Factors and multiples

**Introduction**

**What is a factor?**

Factors are those numbers that divide a given number without leaving any remainder.

**What is a multiple?**

Multiple is defined as the product of one number to another number. If $$a$$ and $$b$$ are two numbers and the product of these two numbers is $$ab$$, then $$ab$$ is said to be the multiple of $$a$$ and $$b$$.

Let us suppose someone drives a car at a constant speed of $$80 \text{km}/ \text{h}$$. If it takes $$10$$ hours to reach the destination, how can we find the distance traveled? It can be obtained using the concept of factors and multiples. The formula to find the distance traveled is $$\text{speed}\times \text{time}$$. So, the distance traveled is $$800\;\text{km}$$. Here, $$800$$ is a multiple of $$80$$ and $$10$$.

**E1.1A:** **Identify and use common factors**

**Common factors**

The common numbers that divide two or more other numbers completely are said to be common factors. For example, two numbers are $$25$$ and $$30$$. Hence, $$1$$ and $$5$$ are the numbers that divide both the numbers completely. So, $$1$$ and $$5$$ are their common factors.

**Highest Common Factor (HCF)**

The highest common factor of two or more numbers is the highest number that divides them completely.

Suppose you have to evaluate the HCF of the numbers $$10$$ and $$15$$. First, write the numbers $$10$$ and $$15$$ as a product of prime numbers.

$$10=2\times 5$$ and $$15=3\times 5$$.

Then write the factors, which are common to both. Here, $$5$$ is the only factor that is common for both. So, the HCF of $$10$$ and $$15$$ is $$5$$.

**Worked examples**

**Example 1**: List all the common factors of the number $$20$$ and $$40$$.

**Step 1: Write the numbers that divide $$20$$ completely.**

The numbers $$1$$, $$2$$, $$4$$, $$5$$, $$10$$, and $$20$$ divide the number $$20$$ completely.

**Step 2: Write the numbers that divide $$40$$ completely.**

The numbers $$1$$, $$2$$, $$4$$, $$5$$, $$10$$, $$20$$, and $$40$$ divide the number $$40$$ completely.

**Step 3: Write the common factors of $$20$$ and $$40$$.**

The common factors of $$20$$ and $$40$$ are $$1$$, $$2$$, $$4$$, $$5$$, $$10$$, and $$20$$.

**Example 2**: Identify the HCF of $$50$$ and $$40$$.

**Step 1: Write the prime factorisation of $$50$$.**

$$50=2\times 5\times\ 5$$

**Step 2: Write the prime factorisation of $$40$$.**

$$40=2\times 2\times 2\times 5$$

**Step 3: Write the common numbers and find their product.**

Common numbers are $$2$$ and $$5$$. Therefore, $$2\times 5=10$$ is the HCF of $$50$$ and $$40$$.

**E1.1B:** **Identify and use common multiples**

**Common multiples**

The numbers that are multiples of two or more numbers are said to be their common multiples. For example, two numbers are $$3$$ and $$4$$. Hence, $$12$$, $$24$$, and $$48$$ are multiples of both numbers. So, these are common multiples of $$3$$ and $$4$$.

**Lowest Common Multiple (LCM)**

The lowest common multiple of two or more numbers is the least number among the common multiples.

Suppose you have to evaluate the LCM of the numbers $$3$$ and $$9$$. First, write the multiple of $$3$$ and $$9$$. The multiples of $$3$$ and $$9$$ are $$3, 6, 9, 12, 15, \cdots$$ and $$9, 18, 27, 36, \cdots$$, respectively. Then, write the lowest multiple that is common for both. Here, $$9$$ is the multiple that is common to both and has the lowest value. So, the LCM of $$3$$ and $$9$$ is $$9$$.

**Worked examples**

**Example 1: **List all the common multiples of the number $$4$$ and $$6$$ that are less than $$30$$.

**Step 1: Write the multiples of $$4$$ that are smaller than $$30$$.**

The numbers $$4$$, $$8$$, $$12$$, $$16$$, $$20$$, $$24$$, and $$28$$ are the multiples of $$4$$.

**Step 2: Write the multiples of $$6$$ that are smaller than $$30$$.**

The numbers $$6$$, $$12$$, $$18$$, and $$24$$ are the multiples of $$6$$.

**Step 3: Write the common multiples of $$4$$ and $$6$$.**

The common multiples of $$4$$ and $$6$$ are $$12$$ and $$24$$.

**Example 2: **Find the LCM of $$4$$, $$6$$, and $$12$$.

**Step 1: Write the multiples of $$4$$.**

The numbers $$4$$, $$8$$, $$12$$, $$16$$, $$20$$, $$24$$, and $$28$$ are the multiples of $$4$$.

**Step 2: Write the multiples of $$6$$.**

The numbers $$6$$, $$12$$, $$18$$, $$24$$, and $$30$$ are the multiples of $$6$$.

**Step 3: Write the multiples of $$12$$.**

The numbers $$12$$, $$24$$, $$36$$, $$48$$, and $$60$$ are the multiples of $$12$$.

**Step 4: Write the lowest number among the multiples of $$4$$, $$6$$, and $$12$$.**

The lowest common multiple of $$4$$, $$6$$, and $$12$$ is $$12$$.

Exam Tip: In the concept of factors, we use the operation ‘division’, and in the concept of multiple, we use the operation ‘multiplication’.