## Factors and Multiples

### Factors

A number that divides a number exactly is called the **factor** of the given number. In other words, an exact divisor of a number is called a **factor** of that number. Example: 2 × 8 =16. Here 2 and 8 divide 16 exactly, hence 2 and 8 are the factors of 16. Similarly, 15 × 3 = 45. Here, 15 and 3 divide 45 exactly, and hence 15 and 3 are the factors of 45, etc.

**Some other Examples**

(a) 6 ÷ 3 = 2

Here, since 3 divides the number 6 exactly, thus 3 is a factor of 6.

(b) 6 ÷ 2 = 3

Here, since 2 divides the number 6 exactly, thus 3 is a factor of 6.

(c) 6 ÷ 1 = 6

Here, since 1 divides the number 6 exactly, thus 1 is a factor of 6.

(d) 6 ÷ 6 = 1

Here, since 6 divides the number 6 exactly, thus 6 is also a factor of 6.

(e) 15 ÷ 5 = 3

Here, since 5 divides the number 15 exactly, thus 5 is also a factor of 15.

(f) 51 ÷ 17 = 3

Here, since 17 divides the number 51 exactly, thus 17 is a factor of 51.

This means multiplicand and multiplicator both are the factors of the product.

### The terms of Multiplication

### Multiplicand, Multiplicator or Multiplier, and Product

The number which is to be multiplied is generally called the **Multiplicand**.

The number by which a number is to be multiplied is called **Multiplicator or Multiplier**.

And the result of multiplication is called the **Product** of the given numbers, that is multiplicand and multiplicator or multiplier.

**Example**

In this example 5 multiplied by 2 gives 10.

Thus, here 5 is multiplicand.

And, 2 is the multiplicator or multiplier.

And, 10 is the product.

### Multiples

When a multiplicand is multiplied by a multiplicator (multiplier), the product is called the multiple of multiplicand or multiplictor (multiplier).

**Example**

5 × 1 = 5

Here, 5, the product is the multiple of 5 or 1.

5 × 2 = 10

Here, 10, the product is the multiple of 5 or 2.

5 × 3 = 15

Here, 15, the product is the multiple of 5 or 3.

7 × 3 = 21

Here, 21, the product is the multiple of 7 or 3.

#### Properties of Factors and Multiples

(a) *A number is always the factors of itself.*

**Example**

5 × 1 = 5

(b) *1 is the factor of every number. *

**Example**

7 × 1 = 7

8 × 1 = 8

9 × 1 = 9

In the above examples, 1 is the factor of 7, 8, and 9.

Since we can get the number after multiplying it by 1, so, it is said that 1 is the factor of every number.

(c) *Every factor of a number is an exact divisor of that number. *

**Example**

20 = 20 × 1

In this example, 20 and 1 are the factors of 20.

And, we can see that 20 and 1 both are exact divisors of 20.

20 = 10 × 2

In this example, 10 and 2 are the factors of 20.

And, we can see that 10 and 2 both are exact divisors of 20.

20 = 5 × 4

In this example, 5 and 4 are the factors of 20.

And, we can see that 5 and 4 both are exact divisors of 20.

20 = 5 × 2 × 2 × 1

Here, 5, 2, and 1 are the factors of 20.

And, 5, 2, and 1 all divide the 20 exactly.

(d) *Every factor is less than or equal to the given number. *

**Example**

50 = 5 × 5 × 2 × 1

In the above example, 5, 2 and 1 are the factors of 50, and are less than 50.

50 = 50 × 1

In the above example, 50 and 1 are the factors of 50. Here 50 is equal to 50 while 1 is less than 50.

(e) *Number of factors of a given number are finite. *

60 = 60 × 1

60 = 2 × 2 × 3 × 5 × 1

In the above examples, the factors of 60 are 60, 1, 2, 2, 3, 5, and 1 total 6.

95 = 19 × 5 × 1

95 = 95 × 1

In the above examples, the factors of 95 are 95, 1, 19, and 5 total 4.

(f) *Every multiple of a number is greater than or equal to that number. *

**Example**

5 × 1 = 5

Here, the multiple of 5 is equal to the 5.

And the multiple of 1 is equal to 5, that is greater than the 1.

5 × 2 = 10

The product is 10 which is the multiple of 5 and is greater than 5.

(g) *The number of multiples of a given number is infinite. *

This means we can get as multiples as we required or desired of a number.

**Example**

2 × 1 = 2

2 × 2 = 4

2 × 3 = 6

2 × 4 = 8 and so on.

## Perfect Number

If the sum of all of the factors of a number is equal to the twice of the number, then the number is called a **Perfect Number**.

## Second definition of a Perfect Number

If the sum of the prime factors of a number is equal to the number itself, then the number is called a **Perfect Number**. [**This definition is applied with the prime factors of a number**]

**Example (a)**

6 = 6 × 1

6 = 3 × 2

Thus, the factors of 6 are 1, 2, 3, and 6, total 4.

And the sum of the factors of 6

= 1 + 2 + 3 + 6 = 12

Here, 12 (the sum of the factors of 6) is equal to 2 × 6. That means the sum of the factors of 6 is twice of 6 which is equal to 12.

Thus, 6 is a perfect number.

**Example (b)**

28 = 28 × 1

28 = 14 × 2

28 = 7 × 4

Thus, factors of 28 are 1, 2, 4, 7, 14, and 28, total 6.

And the sum of the factors of 28

= 1 + 2 + 4 + 7 + 14 + 28 = 56

This means, the sum of the factors of 28 is equal to twice of 28 = 56

Since, the sum of the factors of 28 is equal to the twice of 28 which is equal to 56, thus, 28 is a perfect number.

### Some examples of Perfect Numbers

6, 28, 496, 8128, 33550336, 8589869056 etc. are some examples of **perfect numbers**.