# Euclidean division

Description of the division with remainder and the divisibility relation

## Division without remainder

Euclidean division is the division of two integers, which produces a quotient and a remainder.

If a natural number $$a$$ divides by a natural number $$b$$, then it is calculated how many times the number $$b$$ is contained in $$a$$. The result is the quotient $$q$$ and possibly a remainder $$r$$.

We can write $$a = b · q + r$$

Example   $$17 / 5 = 3$$ remainder $$2$$

The remainder is therefore the difference between the dividend and the largest multiple of the divisor

$$17 - (3 · 5) = 2$$

A remainder arises only if the dividend is not a multiple of the divisor. In other words, if the dividend is not divisible by the divisor.

## Euclidean division and negative numbers

Dividing numbers with different signs gets the following results.

$$7\,/\, 3 = 2$$   Rest   $$1$$

$$-7 \,/\, 3 = -2$$   Rest   $$-1$$

$$7\, / -3 = -2$$   Rest   $$1$$

$$-7\,/ -3 = 2$$   Rest   $$-1$$

## Division without remainder with RedCrab Calculator

The RedCrab Calculator uses the keyword DIV instead of the slash for a division without remainder.

7 div 3=2