# 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.

Division by 1:

When any number is divided by 1, we always get the number itself as the quotient.

For Example:

645 ÷ 1 = 645           7895 ÷ 1 = 7895           85692 ÷ 1 = 85692

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.

Division by itself:

When any number is divided by the number itself, we always get 1 as the quotient.

For Example:

567 ÷ 567 = 1          5478 ÷ 5478 = 1          24768 ÷ 24768 = 1

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

Division of 0 by any number:

When 0 is divided by any number, we always get 0 as the quotient.

For Example:

÷ 953 = 0          ÷ 5759 = 0          0 ÷ 46357 = 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.

Note: The divisor can never be zero. Division by zero is not possible.

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