Problem 108

$\boxed{(E)}$ $2017$
There are $365$ days in a non-leap year. There are $7$ days in a week. Since $365 = 52 \cdot 7 + 1$ (or $365\equiv 1 \pmod{ 7}$), the same date (after February) moves "forward" one day in the subsequent year, if that year is not a leap year.
For example:
$5/27/08$ Tue
$5/27/09$ Wed
However, a leap year has $366$ days, and $366 = 52 \cdot 7 + 2$ . So the same date (after February) moves "forward" two days in the subsequent year, if that year is a leap year.
For example:
$5/27/11$ Fri
$5/27/12$ Sun
You can keep counting forward to find that the first time this date falls on a Saturday is in $2017$:
$5/27/13$ Mon
$5/27/14$ Tue
$5/27/15$ Wed
$5/27/16$ Fri
$5/27/17$ Sat