please help with these question. Thank you

1. Prove that for every integer n 2 0, the sum n3 + (n+ 1)3 + (n+ 2)3 is divisible by 9. 2. Recall Bezout's identity: Let a, b E Z such that a and b are not both zero. Then there exists x, y E Z such that ax + by = god(a, b). Use this result to prove the following result: Let a, b, c E Z such that god(a, b) = 1. Then (a | bc) = (a | c). 3. Let P C N be the set of prime numbers P = {2, 3, 5, 7, 11, ...}. Determine whether the following statements are true or false. Prove your answers ("true" or "false" is not sufficient). 1. Vr EP, Vy EP, xty EP. 2. Vx E P, By E P such that a ty e P. 3. r E P such that, Vy E P, x ty EP. 4. Ex E P such that, By E P, at y e P. 4. Prove the following statement: For every positive number e there is a positive number M such that 2:12 x2 + 1 - 2 > M