Question

If $n$ is any positive integer, then $n^3 - n$ is divisible:

• A. Always by 12
• B. Never by 12
• C. Always by 6
• D. Never by 6

Hint:

When checking divisibility, factor and analyze properties of consecutive integers; they reveal divisibility patterns quickly.

Answer 1

The Correct Option is A

We factor: $n^3 - n = n(n-1)(n+1)$. This is the product of three consecutive integers.
Property 1: In any three consecutive integers, one must be divisible by 3. Therefore, the product is divisible by 3.
Property 2: In any three consecutive integers, at least one is divisible by 2, and another (or the same) is divisible by 4. Therefore, the product is divisible by $2 \times 4 = 8$.
Combining: divisible by 8 and by 3 means divisible by $\mathrm{lcm}(8, 3) = 24$. Since 24 is divisible by 12, the expression is certainly divisible by 12. In fact, it’s stronger: it’s divisible by 24 for all integers $n$.
Thus, it’s “always by 12” is true, but the actual strongest statement is “always by 24.”