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Thread: beginner question

  1. #1
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    beginner question

    if m is an integer show that 3 divides (m3 - m)
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  2. #2
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    Re: beginner question

    Quote Originally Posted by nuebie View Post
    if m is an integer show that 3 divides (m3 - m)
    Proof by induction.
    1) clearly it is true for $m=1$ WHY?

    2) assume it is true for $m=K$ that is $3$ divides $K^3-K$
    Now look at $(K+1)^3-(K+1)=K^3+3K^2+3K+1-K-1=(K^3-K)+3(K^2+K)$ is that divisible by three???
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  3. #3
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    Re: beginner question

    Quote Originally Posted by nuebie View Post
    if m is an integer show that 3 divides (m3 - m)
     m^3 - m \ =

     m(m^2 - 1) \ =

    m(m - 1)(m + 1) \ =

    (m - 1)m(m + 1)


    This is a product of three consecutive integers.

    Consecutive multiples of three differ by three, and one of (m - 1), m, and (m + 1) will be a multiple of three.

    So, that product will be divisible by 3.



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    But if you wanted to, you could make a stronger statement.

    Consecutive multiples of two differ by two, and either one of (m - 1), m, and (m + 1) will be a multiple of two,
    or two of (m - 1), m, and (m + 1) will be a multiple of two, depending on the value of m.

    So, the product (m - 1)m(m + 1) will also be divisible by 2.

    Then the stronger statement can be:

    "If m is an integer, then 6 divides  \ m^3 - m ."
    Last edited by greg1313; Apr 20th 2017 at 03:13 PM.
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