Given below are two statements: Statement 𝐼: 5 divides \(𝑛^5−𝑛\) wherever…
2020
Given below are two statements:
Statement 𝐼: 5 divides \(𝑛^5−𝑛\) wherever \(n\) is a nonnegative integer
Statement 𝐼𝐼: 6 divides \(n^3 - n\) whenever \(n\) is a nonnegative integer
In the light of the above statements, choose the correct answer from the options given below
- A.
Both Statement 𝐼 and Statement 𝐼𝐼 are correct
- B.
Both Statement 𝐼 and Statement 𝐼𝐼 are incorrect
- C.
Statement 𝐼 is correct but Statement 𝐼𝐼 is incorrect
- D.
Statement 𝐼 is incorrect but Statement 𝐼𝐼 is correct
Attempted by 29 students.
Show answer & explanation
Correct answer: A
Answer: Both statements are correct.
Proofs:
Statement I: Since 5 is prime, Fermat's little theorem gives n^5 ≡ n (mod 5) for every integer n. Therefore 5 divides n^5 − n.
Statement II: Factor n^3 − n as n(n−1)(n+1). This is the product of three consecutive integers, so it is divisible by 2 and by 3. Hence it is divisible by 6.
Therefore both statements hold for every nonnegative integer n.