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If (a-1)(b-2) =0, which of the following must be true ?

I. a=1 and b=2
II. if a is not = 1, then b=2
III. if a=1, then b is not=2

(II) is the answer.

For (a-1)(b-2) = 0, a =1 OR b=2

(I) is out because of the and (III) is out because it says if a=1, then b must not be 2, which is wrong. b can still be 0.
(II) satisfiy the logic. If a is not 1, then the only way the equation can go to 0 is for b to be equals to 2.

If a = 1 AND b is any number (1, 2, 100), the equation in the question will still be satisfied. So, Choice 1 is wrong.

Choice 3

It says if a is 1 then b is not 2. If a is 1, b can certainly be 2 and this can be checked by plugging in. So, Choice 3 is wrong.

Choice 2.

It says if a is not 1 then b is 2. This is correct. If a is not 1, and b is not 2, then the equation in the question cannot be satisfied. So if a is not 1, then b HAS to be 2 (which is what Choice 2 says) and if b is not 2 then a HAS to be 1. So Choice 2 is correct and the answer is B.