Mathematics: Let n be a natural number. show that 3 | n3 −n

Let n be a natural number. show that 3 | n3 −n

Wolfgirl2032

09.01.2022

Let

. Then

. By convention, every non-zero integer

divides 0, so $3\vert n^3-n$ .

Suppose this relation holds for n=k

, i.e. $3\vert k^3-k$ . We then hope to show it must also hold for n=k+1

.

You have

(k+1)^3-(k+1)=(k^3+3k^2+3k+1)-(k+1)=(k^3-k)+3(k^2+k)

We assumed that $3\vert k^3-k$ , and it's clear that $3\vert 3(k^2+k)$ because

is a multiple of 3. This means the remainder upon divides (k+1)^3-(k+1)

must be 0, and therefore the relation holds for n=k+1

. This proves the statement.

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chrisholmes176

19.10.2020

My guess would probably be the first one.

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