If your other congruence post is logically prior to this one, then it looks good. Otherwise, you should interpose, twice, the intermediate step AsZ showed in the other post. Just for completeness.
If you rephrase this, the problem is: If does not divide , then also does not divide .
Now, and have the same divisors. So if is not a divisor of , it can't be a divisor of .
This is essentially the same explantion "chiph588@" gave.