2023 · Q37
Consider the following statements :
- 1n³ - n is divisible by 6.2. n⁵ - n is divisible by 5.3. n⁵ - 5n³ + 4n is divisible by 120.
Which of the statements given above are correct?
Elementary MathematicsNumber System
- A1 and 2 only
- B2 and 3 only
- C1 and 3 only
- D1, 2 and 3
Explanation
n³ − n = (n−1)n(n+1) is a product of 3 consecutive integers, hence divisible by 3! = 6. n⁵ − n is divisible by 5 by Fermat's little theorem. n⁵ − 5n³ + 4n = (n−2)(n−1)n(n+1)(n+2) is a product of 5 consecutive integers, hence divisible by 5! = 120. All three statements are correct (D).