Mathematical Reasoning

Step 1: Substitution method. Put x = ω, a cube root of unity.
Step 2: Evaluation. Expression becomes zero, hence divisible by x² + x + 1.

Step 1: Mathematical induction.
This is a problem of mathematical induction. We prove that if is true, then is true.
Thus, the correct answer is (2).

Step 1: Check divisibility by .
Step 2: Using modulo arithmetic, expression is divisible by 64 for all integers .

Step 1: Among four consecutive integers, one is divisible by 4, one by 3, and at least two are even.
Step 2: Hence product is always divisible by .

((2n)!)/((n!)²)=\binom2nn is the central binomial coefficient. It is known that: \binom2nn > (4ⁿ)/(n+1) for n≥ 2 Hence, the given inequality holds for n≥ 2.