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## Mathematicians

Math brain teasers require computations to solve.

 Fun: (2.09) Difficulty: (3.15) Category: Math Submitted By: rajeshwar

There are 4 mathematicians - Brahma, Sachin, Prashant and Nakul - having lunch in a hotel. Suddenly, Brahma thinks of 2 integer numbers greater than 1 and says, "The sum of the numbers is..." and he whispers the sum to Sachin. Then he says, "The product of the numbers is..." and he whispers the product to Prashant. After that, the following conversation takes place :

Sachin : Prashant, I don't think that we know the numbers.
Prashant : Aha! Now I know the numbers.
Sachin : Oh, now I also know the numbers.
Nakul : Now I also know the numbers.

How did they know the numbers?

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 Posted by electronjohn on Feb 11, 2003 Very interesting, although extremely hard to solve. That has got to be one of the longest answers posted so far. My hats off to you for all the work you did on this one. Posted by Codammanus on Feb 12, 2003 I'm not a fan of teasers involving primes. However, the level of thought involved in constructing this one is astonishing. Posted by Gizzer on Feb 18, 2003 He didn't work hard on this one at all. Check out the Mathematicians post under Teasers without answers. Mad-Ade pointed him to another site with this teaser! Posted by smileysteve on Apr 02, 2003 Huh? Posted by gerdmain on Nov 05, 2003 Is this really the only solution? What about a sum of 65? It can be presented as (2,63)...(32,33). All possible products except (4,61) = 244 have multiple answers. Also a sum of 89 and the product 1168(16,73) is another solution, and there are many more. Posted by karanw on Nov 20, 2003 Oh...u cant solve this...! Posted by knbrain on Dec 24, 2004 I got bored reading the answer Posted by Sane on Mar 23, 2005 Posted by Gizzer Feb 18, 2003 He didn't work hard on this one at all. Check out the Mathematicians post under Teasers without answers. Mad-Ade pointed him to another site with this teaser! Owned Posted by yevgen on Sep 02, 2006 In fact, sum 29 = 13+16 is also a solution (the smallest solution above 13+4 = 17). The way to make solution unique is to say that the product is less than 200. Also, the solutoion is very tediously presented. A good way to solve this is the following: 1) the fact that Sachin knew Prashant cannot know the sumbers means that the sum is ODD and also sum munis 2 is composite. Indeed, every even number = sum of two primes (famous Euler's conjecture, easily verified for small numbers). 2) the fact that Prashant knew the answer after that means that his product is uniquely decomposable into two numbers whose sum is ODD. It is easy to verify that this means his product = prime times a power of two (except for 2 itself, since then Prashant would know right away). 3) The fact that Sachin knows as well means that his sum S is uniqulely decomposable as prime+power of two (except for 2 itslef), and also S-2 is not prime. Now, we go through all numbers of the form (composite+2), and check if they are uniquely decomposable as prime+power of 2. This way we see 15 (composite) + 2 = 17 = 13 (prime) + 4 (power of 2), but 17-8 = 9 is composite, and 17-16 is 1 which is disallowed. Going thorough other numbers, the next sum like that is 27 (compsite)+2 = 29 = 13 (prime) + 16 (power of 2), but 29-4=25 (composite) and 29-8 = 21 (composite). However, 13*16=2-8, which is greater than 200, so it can be rules out. Going a bit further we see that next number is indeed 63 (composite) + 2 = 65 = 61 (prime) + 4, since 65-8 = 57 (composite), 65-16=49 (composite), 65-32 = 33 (compsite). Once again, though, 61*4 = 244 > 200, so it's out too. Posted by rachayl on Sep 13, 2006 I'm adding this to my favorites. This is exactly the kind of teaser I was hoping to find when I joined this site. I like thinking about numbers, though I don't have any background in number theory, and solving a prob like this gets me thinking about numbers in all sorts of new ways. Like, I didn't know about Euler's conjecture, but I just spent 20 pleasant minutes conjecturing it on my own! Hope to find more teasers like this!! Thanks! Posted by teasermaster on Dec 19, 2006 I thought this one was unsolvable. Way too hard and there should've been a hint! Posted by teasermaster on Dec 19, 2006 I thought this one was unsolvable. Way too hard and there should've been a hint! Posted by on Feb 02, 2007 To earlier comment - hint probably wouldn't have helped much... Yah this is crazy hard, but interesting none the less Posted by lolliepup on Feb 16, 2007 way...way...too hard. Posted by pating on Jun 25, 2007 What just happened there? I got lost!!! Posted by javlad27 on Jul 28, 2007 That's the longest answer I've ever seen! Posted by sftball_rocks13 on Aug 27, 2007 wow um...the answer was longer than the teaser but...wow i *never* would have gotten that