$n$ being any odd number greater than 1, $n^{65} - n$ is always divisible by

### Concept & Logic This problem utilizes properties of consecutive numbers and factorization. The key insight is that for any odd number $n$, $(n-1)$ and $(n+1)$ are consecutive even numbers, guaranteeing divisibility by higher powers of 2. $$n^{65} - n = n(n^{64} - 1)$$ ### Step-by-Step Solution * **Calculation:** First, factor out $n$ from the expression: $$n^{65} - n = n(n^{64} - 1)$$ * Using the difference of squares, expand $(n^{64} - 1)$ recursively: $$n(n^{32} + 1)(n^{16} + 1)(n^8 + 1)(n^4 + 1)(n^2 + 1)(n - 1)(n + 1)$$ * Look closely at the terms $n(n - 1)(n + 1)$. This can be rearranged as $(n - 1)n(n + 1)$, which is the product of three consecutive integers. * The product of any three consecutive integers is always divisible by $3! = 6$. * Since $n$ is explicitly given as an **odd number**, $(n - 1)$ and $(n + 1)$ are consecutive **even numbers**. * Among any two consecutive even numbers, one is a multiple of 2 and the other is a multiple of 4. Therefore, their product $(n - 1)(n + 1)$ is always divisible by $2 \times 4 = 8$. * Combining these facts: the term $(n-1)n(n+1)$ is divisible by 3 (from being 3 consecutive integers) and divisible by 8 (from the even numbers property). * Since the lowest common multiple of 3 and 8 is 24, the entire expression is always divisible by 24. ### Exam Strategy & Shortcut The fastest method for "always divisible by" questions with variables is **Value Substitution**. * Pick the smallest valid odd number greater than 1: let $n = 3$. * Evaluate the expression for the first few terms: $n(n-1)(n+1) = 3(2)(4) = 24$. * Since 24 must be a factor of the full expression $3^{65}-3$, check the options. 24 is directly present. (Note: While 5 is also a factor due to Fermat's Little Theorem, 24 is the intended answer because it directly relies on the specific "odd number" constraint provided in the prompt). ### Common Pitfall A major pitfall is picking $n=1$ to test, which gives $1^{65} - 1 = 0$, leading to confusion since 0 is divisible by everything. Always read constraints carefully: the question specifies $n > 1$. ### Final Answer Therefore, the correct answer is **24**.