Why is this language over {a,b,c} regular?
All you need to keep track of is the value "number of $a$s seen minus number of $c$s seen", modulo&nbsp;3 &ndash; call that $X$. There are only three different values that $X$ can take. When you see the next character of the input, it's either $a$, $b$ or&nbsp;$c$. Think about how each one affects&nbsp;$X$ and you should be very close to a 3-state automaton that accepts the language you're interested in.