# Long Division with Remainder

For decimal numbers (e.g. 112.45 ÷ 56.7), use : Decimal division.

For polynomials, e.g. (x

^{2}+ 1) ÷ (x - 1), use : Polynomial euclidean division.

## Euclidean division of two positive integers

Doing the integer division of a positive integer (the dividend) by another non-zero positive integer (the divisor) consists on finding two integers called the quotient and the remainder that verifies the equality,

dividend = (quotient × divisor) + remainder

the remainder is an integer less than the divisor.

Example:
17 ÷ 5 = 3 remainder 2

## Euclidean division of two relative integers

The above definition can be generalized to two integers that might be positive or negative (relative integers).

If a is the the dividend and b the divisor,

then there exist unique two integers q (quotient) and r (remainder) such that,

`a = b. q + r` and `0 <= r < |b|`

## Examples

- **Case of positive integers:**

23 ÷ 4 = 5 remainder 3

56 ÷ 7 = 8 remainder 0

- **Cases of relative integers**

-23 ÷ 5 = -5 remainder 2

-65 ÷ 3 = -22 remainder 1

45 ÷ -4 = -11 remainder 1

-26 ÷ -7 = 4 remainder 2

- **Special cases:**

If the dividend is 0 then the quotient and the remainder are equal to 0.

0 ÷ 3 = 0 remainder 0

If the dividend is equal to the dividend then the quotient is equal to 1 and the remainder is 0.

24 ÷ 24 = 1 remainder 0

If the dividend is a multiple of the dividend then the remainder is equal to 0.

9 ÷ 3 = 3 remainder 0

## Integer division and modulo

For two given integers a and b, the remainder of the Euclidean division of a by b is congruent to a modulo b,

`a\equiv r\mod b`

r being the remainder of the Euclidean division of a by b.

## Programming

Here is a program that computes the quotient and the remainder of the Euclidean division of two integer numbers a (dividend) and b (divisor).

Note that in python, the remainder of the integer division may be negative.

In the above calculator, the remainder is always positive (or zero) which guarantees its uniqueness.

### Python

```
def euclidean_division(a, b):
#quotient = a//b, remainder = a % b
return (a//b, a%b)
```

## See also

Long decimal Division

Long Addition

Long Subtraction

Long Multiplication

Modulo operation

Divisibility rules

Divisibility Test