If the sum of five integers is an even integer, what is the greatest number of these integers that can be odd?
A) Zero
B) One
C) Two
D) Four
E) Five

Sums of even and odd numbers work thusly:

even + even = even
odd + odd = even
even + odd = odd

The greatest number of these that can be odd, then, is 4:

(odd + odd) + (odd + odd) + even
= (even) + (even) + even
= even