# 5.21 EuclideanQuotient

`EuclideanQuotient( r, m )`
`EuclideanQuotient( R, r, m )`

In the first form `EuclideanQuotient` returns the Euclidean quotient of the ring elements r and m in their default ring. In the second form `EuclideanQuotient` returns the Euclidean quotient of the ring elements rand m in the ring R. The ring R must be a Euclidean ring (see IsEuclideanRing) otherwise an error is signalled.

A ring R is called a Euclidean ring, if it is an integral ring, and there exists a function delta, called the Euclidean degree, from R-{0_R} to the nonnegative integers, such that for every pair r in R and s in R-{0_R} there exists an element q such that either r - q s = 0_R or delta(r - q s) < delta( s ). The existence of this division with remainder implies that the Euclidean algorithm can be applied to compute a greatest common divisors of two elements, which in turn implies that R is a unique factorization ring. `EuclideanQuotient` returns the quotient q.

```    gap> EuclideanQuotient( 16, 3 );
5
gap> EuclideanQuotient( Integers, 201, 11 );
18 ```

`EuclideanQuotient` calls ```R.operations.EuclideanQuotient( R, r, m )``` and returns the value.

The default function called this way uses `QuotientRemainder` in order to compute the quotient.

GAP 3.4.4
April 1997