Question
Download Solution PDFGiven 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
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFThe 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.
Last updated on Jul 7, 2025
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