Playing with Numbers

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.