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April 30th, 2010, 03:23 PM  #1 
Newbie Joined: Mar 2010 Posts: 13 Thanks: 0  Primes  division
Hello! Determine all primes such that is divisible by . 
May 1st, 2010, 06:32 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,933 Thanks: 2207 
The only even prime (which is 2) has the property described. Suppose some odd prime p has the property, so that p + 8 has such a divisor n. Let q = (p + 8)/n. Clearly q > n, and q and n are both odd (else p = nq  8 would be even). If q = n + 2, p = nq  8 = n(n + 2)  8 = (n  2)(n + 4), but p is a prime, and n = 3 and p = 7 contradicts n² < p, so there is no such solution. If q = n + 4, p = nq  8 = n(n + 4)  8 = (n + 1)² + 2n  9. Since p < (n + 1)², n < 5, and so n = 3 and p = 13. If q = n + 2r (where r ? 3), p = nq  8 = n(n + 2r)  8 = (n + 1)² + (2r  2)n  9, so n = 1, r ? 5 and p = 2r  7, and so r = 5 and p = 3. Hence 2, 3 and 13 are the only primes with the specified property. 
May 1st, 2010, 06:43 PM  #3 
Member Joined: Apr 2010 Posts: 34 Thanks: 0  Re: Primes  division
Notice that you have to take a square root of a prime number. You agree with these statements: 1. If there is an answer, it must be a prime number, and when you plug it into (x+/sqrt(x), you get a whole number. 2. Whole numbers are included in the set of rational numbers, 3. If x is a prime number, (x+/sqrt(x) is irrational when sqrt(x) is irrational. So if sqrt(x) is irrational, x is not a solution. 4. The square root of any number that is not a perfect square is irrational. So we need a number that is both prime and a perfect square. Since a prime number can not have more than two different factors (one and itself), any prime number whose factors are different has the maximum number of factors allowable and can not be a perfect square. Therefore, any prime number whose 2 factors are different will not work. The only prime left is 1, and it satisfies all the conditions. 
May 1st, 2010, 06:49 PM  #4 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Primes  division
Uh, WHAT??? You must not understand the floor function. That's ok; I don't know how to type it! And 1 is not prime. But besides that, everything looks fine. 
May 1st, 2010, 06:51 PM  #5 
Member Joined: Apr 2010 Posts: 34 Thanks: 0  Re: Primes  division
I thought that a prime number is any number that is only divisible by 1 and itself, but that is wrong.

May 1st, 2010, 06:55 PM  #6  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Primes  division Quote:
Other definitions simply state "An integer greater than 2,..." And those brackets around the square root of x mean floor function (i.e. greatest integer)  
May 1st, 2010, 06:56 PM  #7 
Member Joined: Apr 2010 Posts: 34 Thanks: 0  Re: Primes  division
ah, sorry about that

May 1st, 2010, 06:58 PM  #8  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Primes  division Quote:
Check out that function and then look back at skip's answer; it's sufficient and easy to understand.  
May 2nd, 2010, 02:44 AM  #9 
Newbie Joined: Mar 2010 Posts: 13 Thanks: 0  Re: Primes  division
Thanks a lot for help. 

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