We say that three positive integers $x,y,z$ are coprime if there exist three integers $a,b,c$ such that $$ax+by+cz=1$$
How one can prove that $x^{n},y^{m},z^{q}$ are also coprime, i.e., there exist three integers $r,l,w$ such that $$rx^{n}+ly^{m}+wz^{q}=1$$ and what is the relation between $a,b,c$ and $r,l,w$. Here $n,m,q$ are arbitrary.
We have $ax+by+cz = 1$, raise it to the power $m+n+q$ and use the multinomial expansion. $$ \sum_{r+s+t = m+n+q} \binom{m+n+q}{r,s,t} a^rx^rb^sy^sc^tz^t $$
Every term of this summation has either $r \geq n$ or $s \geq m$ or $t \geq q$ (if all three of these were false then $r+s+t < m+n+q$, not possible). So we divide the terms into three parts such that in the first part, $r \geq n$, in the second part ,$s \geq m$ and in the third part $t \geq q$.
The first part is a multiple of $x^n$, the second part is a multiple of $y^m$, the third part is a multiple of $z^q$. The sum of all three parts is $1$, so we have basically got that a sum of a multiple of $x^n$, of $y^m$ and of $z^q$ is $1$, as desired.
I leave you to write out the fine details if you want it, but this is a valid proof for certain.
