17. You can choose four positive integers $X, Y, Z$ and $W$. What is the maximum number of odd numbers you can obtain from the six sums $X+Y, X+Z, X+W, Y+Z, Y+W$ and $Z+W$ ?
A 2
B 3
C 4
D 5
E 6
SolUtion $\quad \mathbf{C}$
The sum of any two even integers is even and the sum of any two odd integers is also even. To obtain an odd number when adding two integers, one must be odd and one must be even. In a set of four integers, if one is odd and three are even there would be three possible sums of two integers that gave an odd number. Similarly, if one is even and three are odd there would also be three possible sums of two integers that gave an odd number. Also, if two of the four integers are odd and two are even, there would be $2 \times 2=4$ possible pairings that gave an odd answer. However, if all four integers are odd or if all four integers are even, there would be no possible sums of two integers that gave an odd answer. Hence the maximum number of odd numbers that could be obtained is 4 .