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 \(-1\)