1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Divisibility Proofs

  1. Jul 8, 2012 #1
    Can anyone help me confirm if I have solved this correctly?

    Many thanks.

    1. The problem statement, all variables and given/known data

    Q. [itex]9^n-5^n[/itex] is divisible by 4, for [itex]n\in\mathbb{N}_0[/itex]

    3. The attempt at a solution

    Step 1: For [itex]n=1[/itex]...
    [itex]9^1-5^1=4[/itex], which can be divided by [itex]4[/itex].
    Therefore, [itex]n=1[/itex] is true...

    Step 2: For [itex]n=k[/itex]...
    Assume [itex]9^k-5^k=4\mathbb{Z}[/itex], where [itex]\mathbb{Z}[/itex] is an integer...1
    Show that [itex]n=k+1[/itex] is true...
    i.e. [itex]9^{k+1}-5^{k+1}[/itex] can be divided by [itex]4[/itex]
    [itex]9^{k+1}-5^{k+1}[/itex] => [itex]9^{k+1}-9\cdot5^k+9\cdot5^k-5^{k+1}[/itex] => [itex]9(9^k-5^k)+5^k(9-5)[/itex] => [itex]9(4\mathbb{Z})+5^k(4)[/itex]...from 1 above => [itex]36\mathbb{Z}+5^k\cdot4[/itex] => [itex]4(9\mathbb{Z}+5^k)[/itex]

    Thus, assuming [itex]n=k[/itex], we can say [itex]n=k+1[/itex] is true & true for [itex]n=2,3,[/itex]... & all [itex]n\in\mathbb{N}_0[/itex]
  2. jcsd
  3. Jul 8, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    That's good. Some remarks on notation:

    Using [itex]\mathbb{Z}[/itex] is not good here. That is the symbol for the set of all integers. You should write [itex]9^k-5^k=4z[/itex] where z is an integer.

    The => should be =
  4. Jul 8, 2012 #3
    Ok. Thank you.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Divisibility Proofs
  1. Divisibility proof (Replies: 5)

  2. Divisibility proof. (Replies: 5)

  3. Proof - Divisibility (Replies: 24)

  4. Divisibility Proof (Replies: 10)

  5. Proof of divisibility (Replies: 3)