Truth In Letters problem

[jerad_bisau]

If a , b, and c are integers and ab^2/c is a positive even integer, which of the following must be true?

I. ab is even
II. ab > 0
III. c is even

I only
II only
I and II
I and III
I, II, and III


The official answer is A. But statement II has to be true since 0 is included, it is correct for negative values, but counter statement is ab <= 0. If ab = 0 then a or b has to be 0 then ab^2 is 0 therefore ab^2/c is 0 and zero is neither negative or positive and problem states that ab^2/c is positive, for that reason II must also be true.

Can you please clarify on this?

[RonPurewal]

The official answer is A. But statement II has to be true since 0 is included, it is correct for negative values, but counter statement is ab <= 0. If ab = 0 then a or b has to be 0 then ab^2 is 0 therefore ab^2/c is 0 and zero is neither negative or positive and problem states that ab^2/c is positive, for that reason II must also be true.

Can you please clarify on this?

sorry, i'm not following this argument.

first, you don't need to consider 0 as a possible value for any of these variables, since that's impossible (given the condition that ab^2/c be a positive even integer). so it's really just an issue of positive vs. negative.

once you've got that out of the way:
realize that you have absolutely no idea whether b is positive or negative. it appears in the expression as b^2, which is always positive (since b is not 0); and therefore, we may merely divide this quantity out to produce the reduced statement: "a/c is positive."
since we have no idea whether b is positive or negative, we do not have a sufficient basis to judge statement II.

if you don't understand this argument, or if you are skeptical, just pick some numbers:
a = 4, b = 1, c = 1 --> these numbers satisfy the condition, and ab > 0
a = 4, b = -1, c = 1 --> these numbers also satisfy the condition, and ab < 0
there you go -- statement II doesn't have to be true.