If $\gcd(a+b,c)=1$ and $a+b+c$ divides $1-abc$, does it follow that $a\mid b$ or $a\mid c$ or $b\mid c$?

Question

Is it true that: For any integers $(\mid a\mid, \mid b \mid, \mid c\mid) \geq 2$ such that $\gcd(a+b,c)=1$, if $a+b+c$ divides $1-abc$ then one of $a$, $b$, and $c$ is a multiple of another ?

Answer 1

If you take $a=5, b=7, c=409$,

then $a+b+c=421$ divides $1-abc=-14314$ and $\text{gcd}(12, 409)=1$;

but neither of $a, b, c$ divides another.

Answer 2

It's not true. Take

$$a = -3,$$ $$b = 2,$$ $$c = 2.$$