It is definitely not necessary. Because the gcd of 2 and 4 is 2, thus we have 2(1) and 2(2). 2 and 1 are coprime and thus will reduce down to 1, the common part

It is definitely not necessary. Because the gcd of 2 and 4 is 2, thus we have 2(1) and 2(2). 2 and 1 are coprime and thus will reduce down to 1, the common part left out is 2 and the uncommon part is 1, as always 1. Thanks about the Buchberger intro I didn't know about that.