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c doesn't have to be even, as b can be 3 and c can be 3, and a is even.

ab>0, not necessarily true. we could have a<0 b>0 and c<0, in which case ab/c >0 but ab<0.

Only ab has to be even, because if it weren't, that means it doesn't have a multiple of 2, and since c is a factor of ab, neither does c.
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From this we know that when the numerator is odd then we will not get a +ve ( even) integer. So the numerator needs to be even. So either a or b need to be even so ab is even.

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

ab^2/c = 2k where k is any +ve integer.

I. ab is even - true: Since 2k is even, either of a or b must be even. Any odd (if ab^2 is odd) divided by any integer (c must be odd) never results in even. So either a or b must be even.

II. ab > 0 - Not true: If a and c are +ve, and b is -ve, 2k is +ve and ab < 0. If a, b and c, all, are -ve, 2k is still +ve and ab> 0.

III. c is even - Not true: If a = 5, b = 4 and c = 1, 2k is even. If a = 4, b = 4 and c = 4, 2k is even. So C is not necessarily an even.