Can someone check my (mathematical) proof?

Question

This is the question:
Let a,b,c be intergers, where a does not equal 0.

If a doesn't divide into b*c with no remainder, then a doesn't divide into b with no remainder and a doesn't divide into c with no remainder.

I used the contrapositive.

Proving that, i said:

Suppose a divides into b and a divides into c.
Thus ax=b and ay=c for intergers x and y.
Thus b*c=(ax)(ay)
b*c=a^2xy
Therefore, a divides into b*c

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If i missed something or if someone can see i got something wrong, please let me know and show me what i should have done, thanks.

Just as a FYI to people who don't know, contrapositive is a certain statement where you take the complete opposite of the given statement and also flip the If and Then parts.

For example, the Contrapositive of "If x=even, then x+1 is odd" would be "If X+1 is Even, then x=odd"

Answer 1

if a doesn't divide b*c it is prime to bc so prime to both b and c as if it were not so it would have divided b or c and so it would have divided bc
you have proved what you have supposed begging the question
you should have started suppose a doesn't' divide b or c then a is prime to bot b and c.so it has to be prime to bc as well

Answer 2

Have you covered the situation where a divides evenly into either b or c, but not both? That's included in the original statement, but I don't think it is in your proof.