Answer: 373

Small Hint:

Each jump changes $x + y$ by $+9$ or $-15$ and changes $x - y$ by $+5$ or $-5$

Big Hint:

Points with $x + y = 3j$ and $x - y = 5k$ need $j$ and $k$ of equal parity, and all of them are reachable. Note $|x| + |y| = \max(|x + y|, |x - y|).$

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. Repeating a jump that changes $k$ in the desired direction first reaches some point on every line $x - y = 5k.$ Two-jump combinations translate by $(9, 9)$ or $(-15, -15).$ Two of the former plus one of the latter give the shift $(3, 3),$ while three of the former plus two of the latter give $(-3, -3),$ so along a fixed line they move $j$ by $2$ or $-2.$ The initially reached value of $j$ has the same parity as $k,$ so these shifts 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 $373.$