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.

3. using items 1 and 2:

(14+4) = 18; only 18 is divided by 3, it means that N=4.

answer: 4.

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