Q8.Marks: +2.0UGC NET Paper 2: Computer Science 2020
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
1.Both Statement I and Statement II are correct✓ Correct
2.Both Statement I and Statement II are incorrect
3.Statement I is correct but Statement II is incorrect
4.Statement I is incorrect but Statement II is correct
Solution
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.