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Guest Message by DevFuse

# More divisibility

11 replies to this topic

### #11 bushindo

bushindo

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Posted 11 March 2013 - 01:05 AM

Indeed, this number is a bit smaller than the one I found.

Spoiler for another perspective

But where is the proof that it is the smallest number that meets  the conditions?

I can't prove that it is the smallest number. The solution that you posed made me realize that my solution is flawed. Originally, I thought I had a divisibility rule for 100 ones, which allowed me to compute the smallest integer within those rules. The number you came up with made me realize that the divisibility rule I had only worked on a subset of the possible divisors.

So, there might be smaller solutions out there. I made a mistake in an earlier post. I should have called that number "the smallest number that I could find" instead of calling it the solution.

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### #12 Prime

Prime

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Posted 12 March 2013 - 02:48 AM

Possibly, you have found the smallest number fitting the conditions in the OP. However, a simple proof/solution does not present itself. When designing new puzzles that comes with the territory.

I appreciate invention of new puzzles. It's a different process from uncovering an answer that has been found before. While playing with this problem I came up with some interesting divisibility rules, which could be used to construct riddles, or break public/private key encryption.

Perhaps, this particular puzzle could be simplified by requiring two digits in the product, rather than three.

Spoiler for divisibility rules

Edited by Prime, 12 March 2013 - 02:56 AM.

• 0

Past prime, actually.

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