Given below are two statements:

Statement I: 5 divides n⁵ - n whenever n is a nonnegative integer.

Statement II: 6 divides n³ - n whenever n is a nonnegative integer.

In the light of the above statements. choose the correct answer from the options given below

The correct answer is option 1

Both statement l and statement ll are correct

Solution:

Statement l: 5 divides n⁵ - n

Using mathematical Induction, if n⁵ - n is divisible by 5, then (n+1)⁵ - (n+1) is also divisible by 5.

If 5 divides, n⁵ - n = 5.k

(n+1)⁵ - (n+1) = n⁵ + 5n⁴ + 10n³ + 10n² + 5n - n

                       = 5.k + 5( n⁴ + 2n³ + 2n² +n)

This shows that (n+1)⁵ - (n+1) is a multiple of 5. Hence it is always divisible by 5.

Statement ll: 6 divides n³ - n

n³ - n = n (n² - 1)

          = n (n + 1)(n - 1)

Here n, n + 1, and n-1 are three consecutive integers. One of them is a multiple of 2 and the other is a multiple of 3. Hence 6 divides n³ - n is always divisible by 6 whenever n is a non-negative integer.