Question

For any positive integer n, prove that n cube - n is divisible by 6​

Answer 1

Step-by-step explanation:

Hi friend!!

→ (n³-n)

Put n = 1,

(1³-1) = 1-1 = 0 is divisible by 6

Put n = 2,

(2³-2) = 8-2 = 6 is divisible by 6

Put n = 3,

(3³-3) = 27-3 = 24 is divisible by 6

Put n = 4,

(4³-4) = 64-4 = 60 is divisible by 6

Therefore,For any positive integer 'n', (n³ -n) is divisible by 6.

Answer 2

Step-by-step explanation:

n³-n

=n(n²-1)

=n(n-1)(n+1)

out of n,n+1,n-1. one of them should be divisible by 2 (since if n is even it is divisible by 2 or if n is odd then n+1 is even and divisible by 2)

the three nos. are also consecutive

thus one of them has to be divisible by 3

since the no. is divisible by 2 and 3 it will also be divisible by 6(divisibility test of 6)