Explanation

gcd(12n+2,8n+1): since 8n+1 is odd, this equals gcd(6n+1,8n+1)=gcd(6n+1,2n)=gcd(6n+1,n)=gcd(1,n)=1 for every n. All 6 values (n=1..6) work. The Euclidean algorithm collapses because 6n+1 leaves remainder 1 on division by n.