If $n$ is an integer , find all the possible values for $(8n+6,6n+3)$ I have got 2 questions which I could not solve:  
1) if $n$ is an integer , find all the possible values for $(8n+6,6n+3)$
2)if $n$ is an integer, find all possible values of $(2n^2+3n+5,n^2+n+1)$

Let $d=\gcd(8n+6,6n+3)$, then $$d\mid 8n+6$$
$$d\mid 6n+3$$ 
so $$d\mid 6(8n+6)-8(6n+3)= 12$$
so $d\in \{1,2,3,4,6,12\}$ Since $6n+3$ is odd $d$ can not be $2,4,6$ or $12$ so $d=1$ or $d=3$ (which is realised at $n=3k$ for some integer $k$)

For second one:
Let $d=\gcd(2n^2+3n+5,n^2 + n+1)$, then $$d\mid 2n^2+3n+5$$
$$d\mid n^2+n+1$$ 
so $$d\mid 2n^2+3n+5-2(n^2 + n+1) =n+3$$
then $$d\mid (n^2+n+1)-(n^2-9)-(n+3)=7$$
So $d=1$ which is ok or $d=7$ which is realised if $n=7k+4$.
