Question

For any positive integer 'n', prove that n²-n is divisible by 6.​

Answer 1

Step-by-step explanation:

Answer:

there are two methods to solve the problem which are discussed below.

Method 1:

Let us consider

a = n3 – n

a = n (n2 – 1)

a = n (n + 1)(n – 1)

Assumtions:

1. Out of three (n – 1), n, (n + 1) one must be even, so a is divisible by 2.

2. (n – 1) , n, (n + 1) are consecutive integers thus as proved a must be divisible by 3.

From (1) and (2) a must be divisible by 2 × 3 = 6

Thus, n³ – n is divisible by 6 for any positive integer n.