Use induction to prove: for every integer n > 1, the number n5 - n is a multiple of 5.

we need to prove : for every integer n>1, the number is a multiple of 5.

1) check divisibility for n=1,  (divisible)

2) Assume that is divisible by 5,

3) Induction,

Now,

Take out the common factor,

     (divisible by 5)

add both the sides by f(k)

We have proved that difference between and is divisible by 5.

so, our assumption in step 2 is correct.

Since is divisible by 5, then must be divisible by 5 since we are taking the sum of 2 terms that are divisible by 5.

Therefore, for every integer n>1, the number is a multiple of 5.