# The largest Monday number

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?

• Can you clarify rule 2 please? Should it state that N contains no more that one of any digit (otherwise digits can appear 3 or more times)? – Gordon K Sep 28 '15 at 8:36
• Why Monday? You named it yourself, or is there such a mathematical concept? – ghosts_in_the_code Sep 28 '15 at 9:03
• My guess is that it is because today is Monday and perhaps he will give us a puzzle every day so tomorrow a puzzle with a Tuesday number might be asked – Ivo Beckers Sep 28 '15 at 9:07
• @corsiKa I would say rule 1 is redundant, not that it doesn't make sense – Kevin Sep 28 '15 at 18:24
• Perhaps I should rephrase "The existence of rule 1 doesn't make sense" – corsiKa Sep 28 '15 at 18:28

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.

• As addition to the answer: We can know that if there is a 7 digit answer that the removed digit must be the digit 4. Because the sum of 98764321 = 40 and the sum of the digits must be divisible by 9 to be divisible by 9 and 4 is the only single digit number that can be removed from 40 to make a number divisible by 9, – Ivo Beckers Sep 28 '15 at 9:00
• That is correct, for 7 digit answers with the number 9 in them. – Fimpellizieri Sep 28 '15 at 9:05
• @Fimpellizieri: and potential seven-digit answers without the number 9 in them are smaller than 9867312 :-) But anyway 9 and 5 can't be the missing digits because of divisibility by 3. – Steve Jessop Sep 28 '15 at 12:22
• @SteveJessop: You meant 9 and 6, right? – Darrel Hoffman Sep 28 '15 at 16:21
• "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." (And if it doesn't have all of the digits 9, 8 and 7, then it can be at most 9864321, which is less than 9867312.) Nice job! – mathmandan Sep 28 '15 at 16:44