A Monday number is a positive integer $N$ with the following three properties:
- The decimal representation of $N$ does not contain the digit 0
- The decimal representation of $N$ does not contain any digit twice
- $N$ is divisible by every digit $D$ that occurs in its decimal representation
What is the largest Monday number?
Answer
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.
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