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#1 2005-08-02 03:03:43

eleusis
Member
Registered: 2005-08-01
Posts: 13

Easy problem

Find three positive whole numbers such that for any two of them, the number one less than their product is divisible by the third number.

This is one of my favorite integer problems.  The wording is a little off.  I think it should read, "their product is evenly divisible", no fractions.  I think there is also a proof that shows there is exactly one solution to this problem.  When posting solutions, please post your method as well.

(x × y – 1)  ÷ z = (int)
(y × z – 1)  ÷ x = (int)
(z × x – 1)  ÷ y = (int)

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#2 2005-08-02 03:40:27

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Easy problem

Does 1, 1 and 1 count?


Why did the vector cross the road?
It wanted to be normal.

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#3 2005-08-02 03:59:58

eleusis
Member
Registered: 2005-08-01
Posts: 13

Re: Easy problem

Ahh, good question.  For this solution, x ≠y ≠z.  For all values of x, y, z such that they are all integers, positive, and unique.

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#4 2005-08-02 06:10:48

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Easy problem

Here's my reasoning, hidden for anyone who wants to do it themselves:


Sorry to anyone uptight about punctuation, but the hide tag has a problem with apostrophes.
Anyway, if you read that, you can see that it was mostly guesswork that got it and we're still nowhere near proving that that's the only combination. I've got as far as showing that there needs to be 1 even and 2 odd, but beyond there I'm stuck.

Last edited by mathsyperson (2005-08-02 06:12:21)


Why did the vector cross the road?
It wanted to be normal.

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#5 2005-08-02 07:58:47

eleusis
Member
Registered: 2005-08-01
Posts: 13

Re: Easy problem

mathsyperson got it.  He used the same method that I did, "plug-and-chug".  Sometimes that's the best way to solve a problem.  I also wrote a program to cycle all combinations of all integers from 1-100,000 and only found one unique solution.  Later, I found on the same website that I originally found the problem a proof showing there was a single unique solution.

I said it was easy.

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#6 2005-08-02 08:01:53

eleusis
Member
Registered: 2005-08-01
Posts: 13

Re: Easy problem

kylekatarn is also correct.  Technically, there are six solutions but they are all permutations of the same three numbers.

I know I am new here but I'm quite impressed with the speed that you found an answer.

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#7 2005-08-02 18:09:14

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,710

Re: Easy problem

mathsyperson wrote:

... but the hide tag has a problem with apostrophes.

I think I got the hide tag straightened out. Quotes should work now


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#8 2005-08-02 19:11:24

ganesh
Administrator
Registered: 2005-06-28
Posts: 35,503

Re: Easy problem

eleusis wrote:

Later, I found on the same website that I originally found the problem a proof showing there was a single unique solution.

Can you post the proof? I got as far as Mathsy did, that two of the numbers should be odd and one even. Maybe, the proof has got something to do with 2 being the only even prime smile


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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