Consider the following statements:<br>1. n^{3}-n is divisible by 6.<br>2. n^{5}-n is divisible by 5.<br>3. n^{5}-5n^{3}+4n is divisible by 120.<br>Which of the statements given above are correct?

Explanation

Statement 1: n^3-n = (n-1)n(n+1), product of 3 consecutive integers, hence divisible by 3! = 6. Statement 2: By Fermat's Little Theorem, n^5-n is divisible by 5. Statement 3: n^5-5n^3+4n = (n-2)(n-1)n(n+1)(n+2) is the product of 5 consecutive integers, divisible by 5! = 120. All statements are correct.

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