5.28 Lcm

Lcm( r1, r2... )
Lcm( R, r1, r2... )

In the first form Lcm returns the least common multiple of the ring elements r1, r2... etc. in their default ring (see DefaultRing). In the second form Lcm returns the least common multiple of the ring elements r1, r2,... etc. in the ring R. R must be a Euclidean ring (see IsEuclideanRing) so that Gcd (see Gcd) can be applied to its elements. Lcm returns the standard associate (see StandardAssociate) of the least common multiples.

A least common multiple of the elements r_1, r_2... etc. of the ring R is an element of smallest Euclidean degree (see EuclideanDegree) that is a multiple of r_1, r_2... etc. We define lcm( r, 0_R ) = lcm( 0_R, r ) = StandardAssociate( r ) and Lcm( 0_R, 0_R ) = 0_R.

Lcm uses the equality lcm(m,n) = m*n / gcd(m,n) (see Gcd).

    gap> Lcm( Integers, 123, 66 );
    2706 

Lcm calls R.operations.Lcm repeatedly, each time passing the result of the previous call and the next argument, and returns the value of the last call.

The default function called this way is RingOps.Lcm, which simply returns the product of r with the quotient of s and the greatest common divisor of r and s. Special categories of rings overlay this default function with more efficient functions.

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