Prove that GCD(x, n + x) = 1 if and only if GCD(x, n) = 1

I'm completely stumped on this proof

You know that gcd of two numbers x,y can be expressed in the form ax+by=gcd.
Apply this:
ax+bn+bx=1--->from the above question
x(a+b)+bn=1---->here a+b is again an integer so i will just take it as k.
kx+bn=1---->k and b are integers.

From the above equation we can say GCD(x,n)=1
Hence Proved :)

for the greatest common denominator to be 1 n and x would have to be prime numbers so basically it would need to be n=7 and x=13 or some other prime number hope that kinda helps