for $111\equiv3(mod 4)$ $222\equiv2(mod 5)$ $333\equiv3(mod 5)$ $555\equiv5(mod 25)$ $777\equiv2(mod 5)$ $888\equiv3(mod 5)$ $999\equiv3(mod 4)$ so only option i $444$ or that $000$ so now we can have 1444 or0000

You forgot $666.$ Also, how about $1444=38^2?$

i edited before $666\equiv2(mod 4)$