Numbers from blackboard

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Fm Jakaria
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Numbers from blackboard

Unread post by Fm Jakaria » Sun Nov 16, 2014 8:10 pm

Consider the following game. In every position of this game, if in a blackboard there are two unequal numbers, remove the least and replace it with the ratio of their product and difference. The game terminates if we get two equal numbers.
Suppose you started with two positive integers in the blackboard and completed the game with finitely many steps.
Prove you get two equal positive integers at last.

Hint:
Pretty induction :D
You cannot say if I fail to recite-
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.

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Phlembac Adib Hasan
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Re: Numbers from blackboard

Unread post by Phlembac Adib Hasan » Mon Nov 17, 2014 1:16 pm

Fm Jakaria wrote:Suppose you started with two positive integers in the blackboard and completed the game with finitely many steps.
Prove you get two equal positive integers at last.
Maybe a typo? Cause, the problem is obvious otherwise. (According to your definitions, the game would not end unless arrived to two equal numbers state.)
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Nirjhor
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Re: Numbers from blackboard

Unread post by Nirjhor » Mon Nov 17, 2014 2:23 pm

Phlembac Adib Hasan wrote:
Fm Jakaria wrote:Suppose you started with two positive integers in the blackboard and completed the game with finitely many steps.
Prove you get two equal positive integers at last.
Maybe a typo? Cause, the problem is obvious otherwise. (According to your definitions, the game would not end unless arrived to two equal numbers state.)
The question asks to prove that you get two equal positive integers at last.
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.


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Phlembac Adib Hasan
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Re: Numbers from blackboard

Unread post by Phlembac Adib Hasan » Tue Nov 18, 2014 12:56 am

Nirjhor wrote:
Phlembac Adib Hasan wrote: Maybe a typo? Cause, the problem is obvious otherwise. (According to your definitions, the game would not end unless arrived to two equal numbers state.)
The question asks to prove that you get two equal positive integers at last.
I didn't notice the 'positive integer' part before. Thanks, anyway.

If the initial numbers are positive integers, then the number replacing rule is actually equivalent to Euclidean algorithm for finding gcd. And The final number should be the lcm of given numbers. (Note: I didn't write "will be" because I haven't rechecked the whole 'fraction-simplifying' part. So it will be great if someone else confirms this.)
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