Question

For any positive integer n, prove that n3 – n is divisible by 8

Answer 1

Explanation:

n³-n = n(n²-1) = n (n-1) (n+1) is divisible by 3 then possible remainder is 0,1 and 2

using the Euclids lemma

let n = 3r , 3r+1 , 3r+2

where r is the integer

case 1 : when n = 3r

then

n³-n = n(n²-1) = n (n-1) (n+1)= 3r(3r-1) (3r+1)

n³-n  is divisible by 3

case 2 : when n = 3r+1

n³-n = 3r+1 (3r) (3r+2) it is divisible by 3

case 3 : when n = 3r-1

n³-n = 3r-1 (3r) (3r-2)it is divisible by 3

thus ,

n³-n  is divisible by 3  for any positive integer

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