11. (6 points) Given that the greatest common divisor of $a$ and $b$ is $4$, the least common multiple of $a$ and $c$, and $b$ and $c$ are both 100, and $a \leqslant$
$b$. How many sets of natural numbers $a$, $b$, and $c$ satisfy these conditions?
【Solution】Solution: According to the problem, we can get that one of $a$ and $b$ is 4, and the other is $4$, $20$, or $100$, so there are 3 possibilities:
(1) $a=4, b=4$, (2) $a=4, b=20$; (3) $a=4, b=100$; For these 3 sets of values of $a$ and $b$, $c$ can be 25, 
50, 100;
Therefore, the natural numbers $a$, $b$, and $c$ that satisfy the above conditions are: $3 \times 3=9$ (sets).
Answer: There are 9 sets of natural numbers $a$, $b$, and $c$ that satisfy the conditions.