If $n\in\Bbb N$, then gcd($8n+1, 7n+1)$ = If  $n\in\Bbb N$, then gcd($8n+1, 7n+1)$ = 
How to do these type of questions?

Suppose that $\gcd(8n+1,7n+1)=d$. Then $d\mid 8n+1$ and $d\mid 7n+1$, so $d\mid n=(8n+1)-(7n+1)$. Then we have $d\mid 1=(7n+1)-7n$. Therefore we must have $d=1$.
The general strategy is to take linear combinations of things you know are divisible by $d$.
