Number Choosing Contest.

RULES:
In order to win, a number must pass five prerequisites and five rules.

A number will not be considered for rule judgement unless it first passes all prerequisites.

The prerequisites (marked α, β, γ, δ, and ε) are ordered by strictness (and should usually be passed unless you are TRYING to break them). The rules (1, 2, 3, 4, 5) are ordered roughly by difficulty, with 1 being the easiest/most obvious and 5 being the most difficult.

And as a hint: there are nine fully-passing numbers below 1000000.