If $c\ge b\ge a$ and $c\mid a^2,b^2$, and $a$ is the minimal number satisfy $c\mid a^2$, does $a$ necessarily divide $b$?

Question

Assume $a$, $b$, $c$ are positive numbers, $a \le b$ and $b \le c$.

If $a^2\equiv0\pmod c$ and $b^2\equiv0\pmod c$, and $a$ is the minimal number satisfy $a^2\equiv0\pmod c$, do we always have $a\mid b$?