Description of the division with remainder and the divisibility relation

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.

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 \(-1\)