Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of odd integers among the 6 integers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
Out of the first two integers, it's possible for both to be even: for example, $10 + 16 = 26.$ But the next two integers, when added, increase the sum by $15,$ which is odd, so one of them must be odd and the other must be even: for example, $3 + 12 = 15.$ Finally, the next two integers increase the sum by $16,$ which is even, so we can have both be even: for example, $2 + 14 = 16.$ Therefore, $\boxed{\textbf{(A) } 1}$ is the minimum number of integers that must be odd.

Sidenote by Williamgolly:
You can use the sum of each pairs of numbers and take mod 2. See [modular arithmetic](https://artofproblemsolving.com/wiki/index.php/Modular_arithmetic)