# Euclidean algorithm (Scala)

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The Euclidean algorithm is an efficient method for computing the greatest common divisor of two natural numbers (or polynomials, or any other object with the necessary structure), and was one of the first known algorithms to be described formally. It is based on the two identities:

*a*>*b*implies: gcd(*a*,*b*) = gcd(*b*,*a*mod*b*)- gcd(
*a*, 0) =*a*

The *gcd* method requires that *a* and *b* are positive integers but doesn't require that *a* > *b*.

<<gcd method>>=defgcd(a:Long, b:Long):Long=(a, b)match{trivial case reversed case base case general case}

The trivial case applies when *a* is equal to *b*:

<<trivial case>>=case(a, `a`)=>a

The reversed case applies when *b* > *a*:

<<reversed case>>=case(a, b)ifb > a=>gcd(b, a)

The base case applies when *b* is equal to 0:

<<base case>>=case(a, 0)=>a

Since *a* > *b* now holds, the first identity can be used for the general case:

<<general case>>=case_=>gcd(b, a % b)

## [edit] Sample code and demonstration

Here's some code demonstrating how to use the above method:

<<Euclid.scala>>=objectEuclid{gcd methoddefmain(args:Array[String]){vala=args(0).toLongvalb=args(1).toLong println(gcd(a, b))}}

If we compile and run the following command lines:

scala Euclid 35 42 scala Euclid 35 40 scala Euclid 35 38

We get the following outputs:

7 5 1

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