2006 AMC 10B Problem 16

1. See how one year moves the weekday. A normal year has $365$ days, and $365 = 52 \cdot 7 + 1$. So after $52$ full weeks there is $1$ day left over, and the same date lands one weekday later.
2. If the year you move through includes a February $29$, it has $366$ days, which is $2$ days past a whole number of weeks. The weekday moves $2$ later.
3. Going from February $29$, $2004$ to February $29$, $2020$ is $16$ one-year steps. List the leap days you pass through: February $29$ of $2008$, $2012$, $2016$ and $2020$. That is $4$ steps that move by $2$, and the other $12$ steps move by $1$.
4. Total shift: $12 \cdot 1 + 4 \cdot 2 = 20$ days forward.
5. Every $7$ days the weekday repeats, so take away whole weeks: $20 - 14 = 6$. Six days after Sunday is Saturday.

The answer is $\boxed{\textbf{(E)}\ \textrm{Saturday}}$.

For weekday questions only the leftover after whole weeks matters, so think of a year as "$1$ extra day" (or $2$ for a leap year) instead of $365$ days. The careful part is counting leap days: count every February $29$ after the starting date, up to and including the ending date.

Count the days directly: $16 \cdot 365 + 4 = 5844$. Now $7 \cdot 835 = 5845$, so $5844$ is one day short of a whole number of weeks. The weekday is one day before Sunday, which is Saturday.

Moving a date forward one ordinary year adds $365 = 52 \cdot 7 + 1$ days, which pushes the weekday forward by $1$. If the year crossed contains a February $29$, the jump is $366$ days and the weekday moves forward by $2$.

From February 29, 2004 to February 29, 2020 is $16$ years. The leap days passed along the way are February 29 of $2008$, $2012$, $2016$, and $2020$: four of them (none of these years is a century exception). So $4$ of the $16$ yearly steps shift by $2$ and the other $12$ shift by $1$:
$$12 \cdot 1 + 4 \cdot 2 = 20 \equiv 6 \pmod 7.$$

Six days after Sunday is Saturday (equivalently, one day back).

The answer is $\boxed{\textbf{(E)}\ \textrm{Saturday}}$.