Relatively prime to c + non-relatively prime to c = relatively prime to c?

Question

If I take a positive integer, a, which is not relatively prime to a positive integer, c, and then add, to a, a positive integer, b, which is relatively prime to c. Is the sum of a+b relatively prime to c?

I thought this up earlier and couldn’t find any answers online.

Here’s a couple examples where this holds:

a=0, b=3, c=10, a+b=3 is relatively prime to 10;

a=6, b=3, c=10, a+b=9 is relatively prime to 10

Ps this is my first post on stack exchange so please let me know if I’ve asked the question incorrectly.

Answer 1

As Andrés showed, the statement does not necessarily hold (counterexample: $a=2,b=3,c=10$).

Some statements that do always hold are:

• $a+b$ is not divisible by $c$.
• $\text{gcd}(a,c)$ is relatively prime to $b$.