This claim is false. Consider a=6, b=3, c=4. Then bc=12, so a divides bc. But a doesn't divide b and a doesn't divide c.
Your answer should contain all three of the following:
The claim is false. Suppose that b and c are primes and a=bc. Then a|bc but a doesn't divide b and also doesn't divide c.
All the math is technically correct, but it's hard for the reader to verify because the counter-example is not concrete. How are you so sure that a doesn't divide b? Disproving a universal claim requires only ONE specific example for which the claim fails. It's more convincing if you are specific.
This claim is false. Consider a=85, b=459, c=190. Then bc=87210, so a divides bc. But a doesn't divide b and a doesn't divide c.
Ok, so this is a bit of a caricature. But subtle versions of this bug are fairly common. Pick a counter-example that is easy for the reader to verify. Technical writing is about effective communication.