Suppose July of year $N$ has five Mondays. Which of the following must occur five times in the August of year $N$? (Note: Both months have $31$ days.)
$\textrm{(A)}\ \text{Monday} \qquad \textrm{(B)}\ \text{Tuesday} \qquad \textrm{(C)}\ \text{Wednesday} \qquad \textrm{(D)}\ \text{Thursday} \qquad \textrm{(E)}\ \text{Friday}$
If there are five Mondays, there are only three possibilities for their dates: $(1,8,15,22,29)$, $(2,9,16,23,30)$, and $(3,10,17,24,31)$. 
In the first case August starts on a Thursday, and there are five Thursdays, Fridays, and Saturdays in August. 
In the second case August starts on a Wednesday, and there are five Wednesdays, Thursdays, and Fridays in August.
In the third case August starts on a Tuesday, and there are five Tuesdays, Wednesdays, and Thursdays in August.
The only day of the week that is guaranteed to appear five times is therefore $\boxed{\textrm{(D)}\ \text{Thursday}}$.