rdevorse wrote:
If a, b, and c are integer and ab + c is odd, which of the must be true?
I. a + c is odd
II. b + c is odd
III. abc is even
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
I understand the reasoning behind the correct answer, but the
MGMAT book's solution also uses a graph which I cannot understand (please see attached file). Can someone explain why under both a and b in the row for two, it is listed as Odd, and then Even for the c column? The book doesn't explain how it sets up the graph.
Thanks
They've listed all possible scenarios for a, b, and c. As each of them can be either even or odd (2 option for each) then there are total of 2*2*2=8 scenarios possible. Then they evaluated each, to see which scenario makes ab+c odd and after that picked these scenarios to work with options I, II, and III.
Personally I'd approach this question in another way.
\(ab+c=odd\), means:
1. \(ab=even\) (which means that
at least one of them is even) and \(c=odd\);
OR:
2. \(ab=odd\) (which means that both are odd) and \(c=even\).
We can see that in both cases at least one is even thus option III (\(abc=even\)) is always true.
Now, for I: if \(c=odd\) (case 1) and \(a=odd\) (so \(b=even\)) then \(a+c=even\neq{odd}\), so this option is not always true.
Exactly the same for option II: if \(c=odd\) (case 1) and \(b=odd\) (so \(a=even\)) then \(b+c=even\neq{odd}\), so this option is not always true.
Answer: C (III only).
Hope it's clear.
_________________