A number is divisible by $3$ if its digits, when added together, are divisible by $3$. For example, take $174: 1 + 7 + 4 = 12$ which is divisible by $3$. You can add it as many times as you want. $12: 1 + 2 = 3$
A number is divisible by $6$ if it is an even number and it is divisible by $3$.
A number is divisible by $4$, if the tens and units form a number which is divisible by $4$, for example $732$ and $9048$ are divisible by $4$ (because $32$ and $48$ are divisible by $4$, but $338$ and $2342$ are not (because $38$ and $42$ are not divisible by $4$). (Why does this work?)
It could be a good idea to make a table to keep track of where the digits $1$ to $6$ could go.
Where will the $5$ have to go?