Final answer:
The expression (n² - 2n)(n² - 1) must be divisible by both 4 and 6 for any integer n greater than 50; however, it cannot be guaranteed to be divisible by 18 without additional information about n.
Step-by-step explanation:
When considering if the expression ((n² - 2n)(n² - 1)) is divisible by 4, 6, or 18 for any integer n greater than 50, we need to analyze the factors of the expression. Let's simplifying the expression component-wise:
- (n² - 2n) can be rewritten as n(n - 2).
- (n² - 1) is a difference of squares, which factors into (n - 1)(n + 1).
Then, we have n(n - 2)(n - 1)(n + 1). This is the product of four consecutive integers, which means it must be divisible by 4 (since among any four consecutive numbers, at least one is divisible by 4, and another is even), and by 6 (since there must be one number divisible by 2 and another by 3).
However, we cannot guarantee it is divisible by 18 (since this would require the presence of both 2 and 9 among our factors), as the specific set of four numbers could, in theory, lack a multiple of 9.
Therefore, the expression must be divisible by both 4 and 6, which corresponds to option (C) I & II only.