(a)

(b)

Assume that xyz is not a multiple of 3. Then x, y, or z must not be a multiple of 3. 

Now, we know that x^3+y^3=z^3. Since x,y,z  is not a multiple of 3,  x^3, y^3, z^3 will have the form 9k+1 or 9k+8.

So, x^3+y^3=z^3 will have the form (9k+1 or 9k+8) + (9k+1 or 9k+8) = (18k+2 or 18k+16 or 18k+ 9) which is not in the form of 9k, 9k+1 or 9k+8.

This contradicts the fact that x^3+y^3=z^3 is a cube of an integer. Therefore, our assumption that xyz is not a multiple of 3 must be false, and xyz must be a multiple of 3.

I don’t think this is helpful, you can refer to Fermat’s last law, which is  “there are no natural numbers (1, 2, 3,…) x, y, and z such that x^n + y^n = z^n , in which n is a natural number greater
than 2”. So x^3+y^3=z^3 in this problem has no integer solution (except x,y, and z are all 0), so it is impossible to prove that xyz is a multiple of 3. Maybe this question is wrong

I changed a simpler way to prove this to you. Could you understand this one?

gcd(a+b, ab)=1, so gcd(a+b, 3ab)=gcd(a+b,3)

If x, y, z are not integers, then xyz can be a decimal, an irrational number or an infinitely repeating decimal, then any number can be a multiple of 3

As I said, there is no integer that can satisfy x^3+y^3=z^3, so this problem cannot be proved, so the first sentence in your proof process is not satisfied. This question should be wrong, because I have seen similar questions but not like this. I suggest you ask your professors for advice. Or you can refer to the proof process of Fermat’s last law, it may help you.

Then I’ll give it a try. Could you send me some of your class materials? I’m not sure what theory you want to use to prove this question.

If the precondition of this question is x^2+y^2=z^2, then it can be proved that xyz is a multiple of 3. I’ve done the proof. But if the precondition is x^3+y^3=z^3, due to the restriction of Fermat’s law, it will definitely not be established. Unless it is pointed out that xyz are all 0 according to Fermat’s last law, so they are multiples of 3, this is the only way to prove it. So I wonder if your teacher made a wrong question. This question can only be established under the premise of x^2+y^2=z^2. Can you ask the teacher about this?

I should not be able to give you any advice, because the conclusion of the first question is not completely correct, so using the conclusion of the first question cannot fully prove the second question. I have modified the proof of the second question, which is not completely correct, but you can refer to it.