This page extends the division in first principles into a simplified procedure for division of integers, which is called *sign property of integer division*.

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The division `14-:3`is understood as

`text(received:)14` is split into `3` equal parts and one part is put-in (positive divisor).

The result of the division is `14-:3= ` quotient `4` and remainder `2`

The division `(-14)-:3`is understood as

`text(given:)14` is split into `3` equal parts and one part is put-in (positive divisor).

The result of the division is `(-14)-:3= ` quotient `-4` and remainder `-2`

The division `14-:(-3)`is understood as

`text(received:)14` is split into `3` equal parts and one part is taken-away (negative divisor).

The result of the division is `14-:(-3)= ` quotient `-4` and remainder `2`

The division `(-14)-:(-3)` is understood as

`text(given:)14` is split into `3` equal parts and one part is taken-away (negative divisor).

The result of the division is `(-14)-:(-3)= ` quotient `4` and remainder `-2`

Summary of integer division illustrative examples:

• `14-:3 = 4text((Q) & ) 2text((R)) ` : `text(received:)14` is split into `3` parts is quotient `text(received:)4` and remainder `text(received:)2`.

• `(-14)-:3 = -4text((Q) & ) -2text((R))` : `text(given:)14` is split into `3` parts is quotient `text(given:)4` and remainder `text(given:)2`.

• `14-:(-3) = -4text((Q) & ) 2text((R))` : `text(received:)14` is split into `-3` parts is quotient `text(given:)4` and remainder `text(received:)2`.

• `(-14)-:(-3) = 4text((Q) & ) -2text((R)) ` : `text(given:)14` is split into `-3` parts is quotient `text(received:)4` and remainder `text(given:)2`.

Based on this, the division is simplified as

• +ve `-:` +ve `=` +ve with +ve remainder

• +ve `-:` -ve `=` -ve with +ve remainder

• -ve `-:` +ve `=` -ve with -ve remainder

• -ve `-:` -ve `=` +ve with -ve remainder

**Integer Division -- Simplified Procedure** : The sign of the quotient and remainder are decided by the signs of dividend and divisor as:**Sign-property of Integer Division**

• positive `-:` positive `=` positive with positive remainder

• positive `-:` negative `=` negative with positive remainder

• negative `-:` positive `=` negative with negative remainder

• negative `-:` negative `=` positive with negative remainder

Sign of the remainder is that of the dividend.

The absolute values of the quotient and remainder are calculated by whole number division of absolute values of dividend and divisor.

*Solved Exercise Problem: *

Find the result of the division `22 -: (-1)`

- `-11`
- `-22`
- `-22`

The answer is "`-22`"

*Solved Exercise Problem: *

Find the result of the division `0-:(-32)`

- `0`
- `-0`
- both the above
- both the above

The answer is "both the above"

*Solved Exercise Problem: *

Find the result of the division `-180 -: (-88)`

- quotient `2` and remainder `4`
- quotient `-2` and remainder `-4`
- quotient `2` and remainder `-4`
- quotient `2` and remainder `-4`

The answer is "quotient `2` and remainder `-4`".

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