Results 1 to 8 of 8

Math Help - Determine the largest prime divisor of 87! + 88!.

  1. #1
    Member
    Joined
    Sep 2006
    Posts
    75

    Determine the largest prime divisor of 87! + 88!.

    Determine the largest prime divisor of 87! + 88!.

    These problems are so hard... (For me anyway.)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    If you have n!+(n+1)! and (n+2) is prime, then (n+2) is the largest prime divisor.

    The next number, 89, is prime, the largest prime divisor of 87!+88! is 89.

    For instance, try 9!+10!, 11 is the largest prime divisor.

    29!+30!...the largest prime divisor is 31.

    This isn't anything I ever tried to prove. The Hackster probably knows it or will figure it out. There's probably already a known theorem relating it.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by galactus View Post
    If you have n!+(n+1)! and (n+2) is prime, then (n+2) is the largest prime divisor.

    The next number, 89, is prime, the largest prime divisor of 87!+88! is 89.

    For instance, try 9!+10!, 11 is the largest prime divisor.

    29!+30!...the largest prime divisor is 31.

    This isn't anything I ever tried to prove. The Hackster probably knows it or will figure it out. There's probably already a known theorem relating it.
    That is the smallest prime divisor not the largest.

    RonL
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    how about 47?. It's the smallest, isn't it?. It's prime and it divides 87!+88!.

    But you're certainly right, Cap'N. I had a major brain fart. I was thinking of prime factors. DUH
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by galactus View Post
    how about 47?. It's the smallest, isn't it?. It's prime and it divides 87!+88!.

    But you're certainly right, Cap'N. I had a major brain fart. I was thinking of prime factors. DUH
    Opps, sorry that's two of us

    RonL
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Jun 2006
    From
    San Diego
    Posts
    101
    87! + 88! = 87! ( 1 + 88 ) = 87!*(89).

    89 is a prime factor.
    The question is, does a higher prime divide 87!

    Suppose p is prime and greater than 89.

    Well 87! = 87*86*85*...*2*1

    Since p doesn't divide 89, then if p divides 87!*(89), then p divides at least one of these terms in 87!.
    This is impossible since p is larger than all terms appearing in the factorial.

    So 89 is the largest prime factor of 87! + 88!

    (this can be worked into a theorem, say if p is prime, then it's the largest prime factor of (p-1)! + (p-2)! )
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    So I was correct. I thought this was something I had seen before, but then second-guessed myself.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,735
    Thanks
    642
    Hello, ceasar!

    Determine the largest prime divisor of 87! + 88!

    We have: . 87! + 88!\:=\:87! + 88\cdot87!

    Factor: . 87!(1 + 88) \:=\:87!(89)


    Therefore, 89 is the largest prime divisor.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: June 19th 2011, 12:56 PM
  2. Find prime divisor
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: May 20th 2010, 09:04 AM
  3. Replies: 2
    Last Post: June 9th 2009, 02:05 AM
  4. Determine largest number:
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 28th 2008, 05:02 AM
  5. Least prime number divisor of 5^3 + 9^3
    Posted in the Algebra Forum
    Replies: 2
    Last Post: April 14th 2007, 06:15 AM

Search Tags


/mathhelpforum @mathhelpforum