What is the largest digit n for which 2345n is divisible by 6?

Consider this option (this is not the only way):
1. according to the condition 2345N is divisible by 6, it means, that N is even number (0;2;4;6 or 8).
2. 6=3*2, then the number 2345N is divisible by 3 also. It means, that the number (2+3+4+5+N)=(14+N) is divisible by 3, where N={0|2|4|6|8}.
3. using items 1 and 2:
(14+4)=18; only 18 is divided by 3, it means that N=4.

4.

Note also, that only 23451; 23454 and 23457 are divided by 3. Only 23454 is divided by 6 (23454/6=3909).