Results 1 to 4 of 4

Math Help - gcd questions (fairly simple)

  1. #1
    Member
    Joined
    Apr 2010
    Posts
    133

    gcd questions (fairly simple)

    I feel like these are stupid questions but I just want to double check myself. (also would i have to show some sort of proof/explanation for my answers?)

    (a) Let a be a positive integer. What is the greatest common divisor of a and a^2? This would be a right? With the idea of if a is a prime number than obviously the only divisor is going to be 1 and itself.

    (b) Let a be a positive integer. What is the greatest common divisor of a and a+2? Would this be 1 if a is odd and 2 if a is even?

    (c) show that if a and be are integers with (a,b)=1, then (a+b,a-b)=1 or 2.
    Last edited by alice8675309; September 29th 2010 at 05:44 PM. Reason: wrong sign
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member MacstersUndead's Avatar
    Joined
    Jan 2009
    Posts
    291
    Thanks
    32
    Quote Originally Posted by alice8675309 View Post
    I feel like these are stupid questions but I just want to double check myself. (also would i have to show some sort of proof/explanation for my answers?)

    (a) Let a be a positive integer. What is the greatest common divisor of a and a^2? This would be a right? With the idea of if a is a prime number than obviously the only divisor is going to be 1 and itself.

    (b) Let a be a positive integer. What is the greatest common divisor of a and a+2? Would this be 1 if a is odd and 2 if a is even?

    (c) show that if a and be are integers with (a,b)=1, then (a+b,a-b)=1 or 2.
    I don't think these are stupid questions; I myself would like to know the solutions of these for practice.

    For (a), do you know the fact that every integer can be expressed as a product of primes, and the prime factorization of the gcf?

    ie a = \prod p^{\alpha_p}

    and that  gcf(a,b) = \prod p^{min \alpha_p , \beta_p} ?

    In this case, min{\alpha_p, \alpha_p^2} = \alpha_p which shows the gcf(a, a^2) = a.
    --

    For (b) and (c), by inspection, I would most likely recourse to integer sums. They look tricky though.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Quote Originally Posted by alice8675309 View Post
    I feel like these are stupid questions but I just want to double check myself. (also would i have to show some sort of proof/explanation for my answers?)

    (a) Let a be a positive integer. What is the greatest common divisor of a and a^2? This would be a right? With the idea of if a is a prime number than obviously the only divisor is going to be 1 and itself.

    (b) Let a be a positive integer. What is the greatest common divisor of a and a+2? Would this be 1 if a is odd and 2 if a is even?

    (c) show that if a and be are integers with (a,b)=1, then (a+b,a-b)=1 or 2.
    I agree with your answers for (a) and (b), here is some justification.

    (a) a^2 = a*a so clearly a divides a^2, and also a=1*a so a divides a (which you probably can use without thinking by this point). It should also be obvious that any integer greater than a cannot divide a. (You can prove this rigorously if you wish, it's easy.) Therefore gcd(a,a^2)=a.

    (b) You should know in general that if d | x and d | y then d | (x-y) and d | (x+y). So gcd(a,a+2) divides 2. So it is either 1 or 2. You can then easily conclude that if a is even then gcd is 2, else it is 1.

    (c) I think this is fairly tricky. Let g = gcd(a+b,a-b). We know from what I wrote in (b) that g | 2a and g | 2b. Can you take it from here?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Apr 2009
    Posts
    678
    Thanks
    1
    infact for (c) you can find out when would it be 1 or 2 and this depends on whether a,b are even/odd
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: September 24th 2010, 06:47 AM
  2. Replies: 0
    Last Post: September 23rd 2010, 11:35 AM
  3. Fairly simple limit proof?
    Posted in the Calculus Forum
    Replies: 7
    Last Post: June 27th 2010, 06:28 PM
  4. Help with fairly simple, quick problem.
    Posted in the Algebra Forum
    Replies: 2
    Last Post: August 26th 2008, 06:44 PM
  5. I need help with a fairly simple question
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 29th 2008, 03:52 AM

Search Tags


/mathhelpforum @mathhelpforum