Euclidean division

Description of the division with remainder and the divisibility relation

Division mit Rest


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   \(-1\)