Dozens Poster

54312

Why?

It must be a multiple of 3

1 + 3 + 4 + 5 = 13 so the remaining digit could be 2, 5, 8

Try 8. It must be a multiple of 4 so the last two digits must be divisible by 4.

85314

2-digit multiples of 4 which end in 4 (or 8) have second-last digit even: 04, 24, 44, 64, 84

Best option with 8: 53184

Try 5: impossible since we can't make a multiple of 4

Try 2. 

Using previous pattern, 53124

But 2-digit multiples of 4 which end in 2 have second-last digit odd: 12, 32, 52, ...

54312

This is the largest possibility