If $${b_1}{b_2} = 2({c_1} + {c_2})$$ and b₁, b₂, c₁, c₂ are all real numbers, then at least one of the equations $${x^2} + {b_1}x + {c_1} = 0$$ and $${x^2} + {b_2}x + {c_2} = 0$$ has

The greatest integer which divides $$(p + 1)(p + 2)(p + 3)...(p + q)$$ for all $$p \in N$$ and fixed $$q \in N$$ is

If p, q are odd integers, then the roots of the equation $$2p{x^2} + (2p + q)x + q = 0$$ are

Let $$\alpha$$ and $$\beta$$ be the roots of $${x^2} + x + 1 = 0$$. If n be a positive integer, then $$\alpha$$ⁿ + $$\beta$$ⁿ is