Notice 9867312 is a Monday number.
The largest Monday number may not contain 5 because in this case it would end in 5, and thus not be divisible by 2, 4 and 8, so it would have at most 6 digits.
On the other hand, a Monday number may not have 8 digits. Indeed, if that were the case, the preceding paragrph would imply such a number has each digit but 0 and 5 in it. In particular, it would have the digit 3. But the sum of its digits would be 1 + 2 + 3 + 4 + 6 + 7 + 8 + 9 = 40, which is not divisible by 3.
It follows that the largest Monday number must have 7 digits. If it has the digits 9, 8 and 7 it must be a multiple of 504, and it's easy check the highest Monday number that is a multiple of 504 is 9867312. Because we know the largest Monday number has 7 digits, it follows that this is the largest such number.