Properties of Division

The properties of division of whole numbers are as follows :

Property 1:

If a and b (b not equal to zero) are whole numbers, then a ÷ b (expressed as a/b) is not necessarily a whole number.

In other words, whole numbers are not closed for division.

Verification: We know that dividing a whole number a by a non-zero whole number b means finding a whole numbers c such that a = bc.

Consider the division of 14 by 3. We find that there is no whole number which when multiplied by 3 gives us 14. So, 14 ÷ 3 is not a whole number. Similarly, 12, 5, 9, 4, 37, 6 etc. are not whole numbers.

Property 2:

If a is any whole number, then a ÷ 1 = a.

In other words, any whole number divided by 1 gives the quotient as the number itself.

Verification: We know that

(i) 1 × 5 = 5

Therefore, 5 ÷ 1 = 5

(ii) 1 × 11 = 11

Therefore, 11 ÷ 1 = 11

(iii) 1 × 29 = 29

Therefore, 29 ÷ 1 = 29

(iv) 1 × 116 = 116

Therefore, 116 ÷ 1 = 116

(v) 1 × 101 = 101

Therefore, 101 ÷ 1 = 101

(vi) 1 × 1 = 1

Therefore, 1 ÷ 1 = 1

Property 3:

If a is any whole number other than zero, then a ÷ a = 1.

In other words, any whole number (other than zero) divided by itself gives 1 as the quotient.

Verification: We have,

(i) 13 = 13 × 1

Therefore, 13 ÷ 13 = 1

(ii) 9 = 9 × 1

Therefore, 9 ÷ 9 = 1

(iii) 17 = 17 × 1

Therefore, 17 ÷ 17 = 1

(iv) 123 = 123 × 1

Therefore, 123 ÷ 123 = 1

(v) 21 = 21 × 1

Therefore, 21 ÷ 21 = 1

(vi) 1 = 1 × 1

Therefore, 1 ÷ 1 = 1

Property 4:

Zero divided by any whole number (other than zero) gives the quotient as zero. In other words, if a is a whole numbers other than zero, then 0 ÷ a = 0

Verification : We have,

(i) 0 × 7 = 0

Therefore, 0 ÷ 7 = 0

(ii) 0 × 11 = 0

Therefore, 0 ÷ 11 = 0

(iii) 0 × 17 = 0

Therefore, 0 ÷ 17 = 0

(iv) 0 × 132 = 0

Therefore, 0 ÷ 132 = 0

(v) 0 × 164 = 0

Therefore, 0 ÷ 164 = 0

Note:

In order to divide 6 by 0, we must find a whole number which when multiplied by 0 gives us 6. Clearly, no such number can be obtained. We, therefore, say that division by 0 is not defined.

Property 5:

Let a, b and c is the whole numbers and b ≠ 0, c ≠ 0. If a ÷ b = c, then b × c = a.

Verification: We have,

(i) 15 ÷ 3 = 5

Therefore, 5 × 3 = 15

(ii) 27 ÷ 9 = 3

Therefore, 9 × 3 = 27

(iii) 56 ÷ 7 = 8

Therefore, 7 × 8 = 56

(iv) 99 ÷ 11 = 9

Therefore, 11 × 9 = 99

(i) 75 ÷ 15 = 5

Therefore, 15 × 5 = 75

Property 6:

Let a, b and c be whole numbers and b ≠ 0, c ≠ 0. If b × c = a, then a ÷ c = b and a ÷ b = c.

Verification: We have,

(i) 18 = 3 × 6

Therefore, 16 ÷ 3 = 8 and 16 ÷ 8 = 3

(ii) 42 = 6 × 7

Therefore, 42 ÷ 6 = 7 and 42 ÷ 7 = 6

(iii) 72 = 8 × 9

Therefore, 72 ÷ 9 = 8 and 72 ÷ 8 = 9

(iv) 48 = 8 × 6

Therefore, 48 ÷ 6 = 8 and 48 ÷ 8 = 6

(v) 24 = 12 × 2

Therefore, 24 ÷ 2 = 12 and 24 ÷ 12 = 2

Property 7:

(Division Algorithm) If a whole number a is divided by a non-zero whole number b, then there exists whole numbers q and r such that a = bq + r, where either r = 0 or, r < b.

This can also be expressed as:

Related to this we have the following definitions:

DIVIDENDThe number which is to be divided is called dividend.In this case, a is the dividend.

DIVISOR: Divisor is the number by which the dividend is divided.Here, b is the divisor.

QUOTIENT: The number of times the divisor divides the dividend is called the quotient.

Here, q is the quotient.

REMAINDER: The number which is left over after division is called the remainder.

Here, r is the remainder. Clearly r = a – bq

Using these terms, the division algorithm can be restated as:

Dividend = Divisor × Quotient + Remainder.

Verification: Let a = 159 and b = 8.

By actual division, we find that

Quotient (q) = 19

And, Remainder (r) = 7

Clearly, 159 = 19 × 8 + 7 i.e. a = bq + r.

The Number Zero

Properties of Whole Numbers

Successor and Predecessor

Representation of Whole Numbers on Number Line

Properties of Subtraction

Properties of Multiplication

Properties of Division

Division as The Inverse of Multiplication