Results 1 to 3 of 3

Math Help - Number theory, prime numbers, interesting problem

  1. #1
    Junior Member
    Joined
    Sep 2007
    Posts
    37

    Number theory, prime numbers, interesting problem

    Tell me does this problem have any solution? I cannot find any.... :blush: But I think this question is very interesting and have nice solution, help me:

    numbers are given:

    (1) ab + bc + ac

    (2) a^3 + b^3 + c^3 - 2abc

    Numbers a, b, c are prime numbers and we have to find what numbers a, b, c that (1) and (2) are divisible by (a + b + c).

    So the question is: Find all (a, b, c) [a, b, c - prime numbers], that (1) and (2) are divisible by (a + b + c).

    I am very sorry for my english. Can anybody help me?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Ununuquantium View Post
    Tell me does this problem have any solution? I cannot find any.... :blush: But I think this question is very interesting and have nice solution, help me:

    numbers are given:

    (1) ab + bc + ac

    (2) a^3 + b^3 + c^3 - 2abc

    Numbers a, b, c are prime numbers and we have to find what numbers a, b, c that (1) and (2) are divisible by (a + b + c).

    So the question is: Find all (a, b, c) [a, b, c - prime numbers], that (1) and (2) are divisible by (a + b + c).

    I am very sorry for my english. Can anybody help me?
    Let d = a + b + c. Then

    d(bc+ca+ab) = (b^2c+c^2a+a^2b) + (bc^2+ca^2+ab^2) + 3abc,

    d^3 = (a^3+b^3+c^3) + 3(b^2c+c^2a+a^2b) + 3(bc^2+ca^2+ab^2) + 6abc,

    and therefore

    d^3 -3d(bc+ca+ab) = (a^3+b^3+c^3) -3abc = \{(a^3+b^3+c^3) -2abc\} - abc.

    If d divides (a^3+b^3+c^3) -2abc then it follows that d divides abc. Since a, b and c are primes, and d is clearly greater than any of them, it follows that d must be equal to one of the products bc, ca, ab or abc. It is fairly easy to deduce from this that the only solution to the problem is a=b=c=3.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2007
    Posts
    14
    but for example a=2, b=3, c=5 ? (2*3*5) is divided by (2+3+5)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prime Number Theory
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: June 13th 2009, 08:17 PM
  2. Extremely interesting number theory problem?
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: November 1st 2008, 07:12 PM
  3. Number Theory relatively prime.
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: July 9th 2008, 07:04 PM
  4. Number Theory - Prime divisors
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: August 10th 2007, 06:05 AM
  5. number theory, prime
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 17th 2006, 05:10 PM

Search Tags


/mathhelpforum @mathhelpforum