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

Let . Then . By convention, every non-zero integer divides 0, so .

Suppose this relation holds for , i.e. . We then hope to show it must also hold for .

You have



We assumed that , and it's clear that because is a multiple of 3. This means the remainder upon divides must be 0, and therefore the relation holds for . This proves the statement.