Answer: 373

Solution:

Each jump changes $x + y$ by $+9$ or $-15$ and changes $x - y$ by $\pm 5.$ Starting from $(0, 0),$ every reachable point therefore has $x + y = 3j$ and $x - y = 5k$ for integers $j$ and $k;$ moreover $x = \frac{3j + 5k}{2}$ must be an integer, so $j$ and $k$ have the same parity. Since $|x| + |y| = \max(|x + y|,\, |x - y|),$ the condition $|x| + |y| \le 100$ becomes $|j| \le 33$ and $|k| \le 20.$

Conversely, every such point is reachable: a single jump moves between neighboring lines $x - y = 5k$ (changing $j$ by $3$ or $-5,$ which flips its parity), and two-jump combinations translate by $(9, 9)$ or $(-15, -15),$ which combine — two of the former plus one of the latter — into the shift $(3, 3),$ moving $j$ by $2$ along a fixed line. Together these reach every pair $(j, k)$ of equal parity.

Counting: even $j$ ($33$ values) pairs with even $k$ ($21$ values), and odd $j$ ($34$ values) with odd $k$ ($20$ values), so $M = 33 \cdot 21 + 34 \cdot 20 = 1373.$ The remainder is $373.$