In year $N$, the $300^{\text{th}}$ day of the year is a Tuesday. In year $N+1$, the $200^{\text{th}}$ day is also a Tuesday. On what day of the week did the $100$th day of year $N-1$ occur?
$\text {(A)}\ \text{Thursday} \qquad \text {(B)}\ \text{Friday}\qquad \text {(C)}\ \text{Saturday}\qquad \text {(D)}\ \text{Sunday}\qquad \text {(E)}\ \text{Monday}$
There are either \[65 + 200 = 265\] or \[66 + 200 = 266\] days between the first two dates depending upon whether or not year $N$ is a leap year. Since $7$ divides into $266$ but not $265$, for both days to be a Tuesday, year $N$ must be a leap year.
Hence, year $N-1$ is not a leap year, and so since there are \[265 + 300 = 565\] days between the date in years $N,\text{ }N-1$, this leaves a remainder of $5$ upon division by $7$. Since we are subtracting days, we count 5 days before Tuesday, which gives us  $\boxed{\mathbf{(A)} \ \text{Thursday}.}$