100pts, brainliest: Prove for every positive integer n that (n^3 - n) is divisible by 3.

Proof by Induction

Step-by-step explanation:

n³ - n

Proof by induction

Let n = 1

1³ - 1 = 1 - 1 = 0

0 is a multiple of 3

Assume true for n = k

k³ - k is a multiple of 3

Prove for n = k + 1

(k + 1) ³ - (k + 1)

k³ + 3k² + 3k + 1 - k - 1

k³ + 3k² + 2k

k³ + 3k² + 3k - k

(k³ - k) + 3k (k - 1)

Is a multiple of 3 because:

k³ - k was assumed

3k (k - 1) has a factor '3'

Hence the sum is also a multiple of 3

Or n=1, we have 1-1=0, 0 is divisible by 3.

Assume that this is true for n and we have to show that its true for n+1

(n+1) 3 - (n+1) = n3+3n2+2n=n3-n+3 (n2+n) is divisible by 3; n3-n by 3 by is divisible by assumption and 3 (n2+n) is divisible as well. So, if n3-n is divisible by 3, then (n+1) 3 - (n+1) is divisible as well.