1. ## beginner question

if m is an integer show that 3 divides (m3 - m)

2. ## Re: beginner question

Originally Posted by nuebie
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???

3. ## Re: beginner question

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

$\displaystyle m(m^2 - 1) \ =$

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

$\displaystyle (m - 1)m(m + 1)$

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 $\displaystyle \ m^3 - m$."