Integers and Diophantine Equations

Question

Assume that $n,m,p\in\mathbb{Z}$ and $\gcd(nm,p)=1$. Show that $\gcd(m,p)=1$. For this question I don't think the $n$ factor effects the $\gcd$, but I don't know how to prove this mathematically (proofs are hard for me).

Answer 1

$gcd(mn,p)=1$ so there exists integers $u$ and $v$ such that $umn+pv=1$, hence $(un)m+pv=1$, hence since $gcd(m,p)$ divides $(un)m+pv=1$, then it is $1$.

Answer 2

$\gcd(mn,p) = 1$ means the system $$ mn = q_1 d \quad p = q_2 d $$ has only the solution $d = 1$ for integer $q_1$, $q_2$. If $\gcd(m,p) = d' > 1$ it would solve $$ m = q_1' d' \quad p = q_2' d' $$ which would solve $$ mn = (q_1' n) d' $$ as well, which would mean $d'$ would be a common divisor of $mn$ and $p$ as well. But we already know that there is only $1$ as common divisor. So our assumption $d'>1$ must be wrong and we are left with $1=d'=\gcd(m,p)$.