Problem Solving

If abc = b^3, which of the following must be true?

I. ac = b^2
II. b = 0
III. ac = 1

A. None
B. I only
C. II only
D. I and III
E. II and III

[spoiler]OA=A[/spoiler].

I don't understand this PS question. Why is A the correct choice? Can anyone explain this to me? Please.

Try some numbers

Let B=0, then A*0*C = 0

Test the first possibility: Does A*C have to equal B^2= 0 for the above to be true ? No, A and C could be any number

Second possibility can't be ruled out since we assumed it above

Third possibility: Does A*C = 1. Not necessarily, per the first possibility

Now, try some new numbers to see if B=0 is required

Let A=1, B=1 and C=1. Do these multiplied together = B^3 =1 ? Yes. Is B=0 ?, No. Therefore the second possibility, B=0 isn't necessarily true

Answer None, A

$$abc=b^3$$
$$b\left(ac-b^2\right)=0$$
$$So,\ it\ is\ either\ b=0,\ or\ ac=b^2$$
Therefore, NONE of the given option is true
if b=0, then 'ac' will be equal to any number which makes option I and III very uncertain.

Also, If 'ac=b^2' then 'b' can also be any number so Option II is also uncertain.