Is the product of a and b equal to 1?

(1) a∗b∗a=a

(2) b∗a∗b=b

OA:

Question: is ab=1?

(1) a^2*b=a → a^2*b−a=0 → a(ab−1)=0 → either a=0 (and b=any value, including zero) so in this case ab=0≠1 OR ab=1. Two different answers, not sufficient.

(2) a*b^2=b → a*b^2−b=0 → b(ab−1)=0 → either b=0 (and a=any value, including zero) so in this case ab=0≠1 OR ab=1. Two different answers, not sufficient.

(1)+(2) either a=b=0, so in this case ab=0≠1 and the answer to the question is NO, OR ab=1 and the answer to the question is YES. Two different answers, not sufficient.

I had figured that Each alone is insufficient.

But if we have two statements with only one common solution, then shouldnt that be the answer?

from 1 we get: either a=0 or ab=1

from 2 we get: either b=0 or ab=1

combining both we get only one common option, ab=1 - Sufficient