# Properties of Dividing Integers

The properties of dividing integers are discussed here along with the examples.

1. If ‘a’ and ‘b’ are any two integers, then ‘a’ ÷ ‘b’ is not necessarily an integer.

For example:

(i) +12/+3 = +4, which is an integer.

(ii) +45/-15 = -3 which is an integer.

(iii) -135/+9 = -15 which is an integer.

(iv) -725/-25 = + 29 which is an integer.

But,

(v) (+7)/(+4) is not an integer and same is true for (-5) ÷ (+2), (+15) ÷ (-7), (-10) ÷ (-3), etc.

2. If ‘a’ is not negative integer i.e., a ≠ 0; then ‘a ÷ a’ is always equal to unity (1).

For example:

(i) (-3) ÷ (-3) = (+1) = 1

(ii) (+9) ÷ (+9) = (+1) = 1

(iii) (+17) ÷ (+17) = (+1) = 1

(iv) (-25) ÷ (-25) = (+1) = 1                            and so on.

3. For any non-zero integer ‘a’, 0 ÷ a = 0, but a ÷ 0 is not defined.

When zero (0) is divided by any non-zero number, the result (quotient) is always zero and when any number is divided by zero (0), the result is not-defined.

i.e., Zero/Any non-zero number = Zero       and       Any number/Zero = Not-defined

For example:

(i) 0/12 = 0, 0/(-15) = 0, 0/123 = 0                            and so on.

(ii) 15/0 = not-defined, -18/0 = not-defined, 0/0 = not-defined.

Similarly, 0 ÷ 7 = 0, 0 ÷ (-10) = 0, but 12 ÷ 0 is not defined and so is (-15) ÷ 0 and so on.

Also, a ÷ b ≠ b ÷ a

For example:

4 ÷ 2 ≠ 2 ÷ 4

a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c

For example:

8 ÷ (4 ÷ 2) ≠ (8 ÷ 4) ÷ 2                            and so on.