Solving gcd problem , is $\gcd(x,y)=1$ equivalent to $ p \not \mid x $ for EVERY prime divisor of $y$

Question

Given two integers $x$ and $y$,

Is $\gcd(x,y)=1$ equivalent to $ p \not \mid x $ for EVERY prime divisor of $y$

the actual question is to find all integers $k$ such that

$$\gcd(k+8,18)=1$$

thanks.

Answer 1

Notice that if $k$ satisfies the condition, that is, $\gcd(k+8,18)=1,$ then since $\gcd((k+18)+8,18)=1,$ $k+18$ also satisfies the condition.

Thus, you only need to consider solutions to this equation for $0\le k<18,$ and every other solution is a multiple of $18$ plus one of these "base" solutions.