Crazy 7-digit number: all diff. digits, is multiple of each

[alphabetagamma]

A seven digit number has all digits different and is a multiple of each of these digits. What are the seven digits?

I can only get it down to 4+4! possible combinations

Good luck

 

[Denis]

7 digits are 1,2,3,6,7,8,9

There are 105 combos, from 1289736 to 9867312

 

[alphabetagamma]

How do you eliminate all other possible digit combinations like 1,2,3,4,6,7,9 or 1,2,3,4,6,8,9 or any other combinations that seem moderately reasonable. Did you use a computer program?

 

[mmm4444bot]

How do you eliminate ... combinations like 1,2,3,4,6,7,9 ...

Numbers that end in 9 are not divisible by 2.

 

[Denis]

How do you eliminate all other possible digit combinations like 1,2,3,4,6,7,9 or 1,2,3,4,6,8,9 or any other combinations that seem moderately reasonable. Did you use a computer program?

Yes, wrote short program to "get 'em".
But had few short cuts, like 7digit number MUST be even, since there will be at least two even digits.

I don't know of a way to determine that digits 4 and 5 are excluded;
perhaps someone who studied number theory can...
But anyone who tries to get the actual numbers by hand needs to get a life :wink:

 

[alphabetagamma]

Re:

How do you eliminate ... combinations like 1,2,3,4,6,7,9 ...

Numbers that end in 9 are not divisible by 2.

I didn't mean to use that as one of the numbers that is divisible by each digit, I was just saying that is a reasonable combination of numbers, when rearranged it could yield some number that is even.

Without a computer program to do this for me. I think it is true to say that the multiple of all odd numbers must not be a number that ends in 5 or 0. Obviously 0 cannot be in the digit combination. So that gets it down to the 4 even numbers paired with: 3,7,9 or 1,3,7 or 1,3,9 or 1,7,9. And also it could be 1,3,7,9 and any 3 even numbers. That is as far as I could get with thoughts, anyone see how to eliminate anymore? With just thinking and no computer.

The question only requires the naming of the 7 digits, and not how many ways you can arrange them, that would be a waste of time.

 

[mmm4444bot]

I didn't mean to use that as one of the numbers ...

Ah, I misread your question to be about combinations rather than about digits. Please excuse me.

~ Mark