Please give step by step answers.

This is mathematical reasoning class

Thank you!

(1) Prove that for each a: E Z the product 23(3: + 3) is an even integer. (2) Prove by contradiction that no two consecutive integers can both be odd. (3) Prove by contradiction that 2J5 is irrational. (4) Prove that the sum of 5 consecutive integers is divisible by 5. (5) Prove that for a, b, c E Z {0}, if a2 + b2 = c2 then a and 6 cannot both be odd. (6) Prove that there is no positive integer p such that the numbers p, p + 8, p + 22 are all prime. (7) For the following statements, write in symbolic language, determine if true or false (justifying with a proof or counterexample) and negate the statement: (a) 'Every integer is greater than 4 or less than 6.' (b) 'If an integer is both square and prime then it is negative.\" (c) 'If the product of a rational number :1: and a non-zero real number 3; is rational, then :1: is zero or y is rational'. ((1) There are two distinct rational numbers that do not have another rational number strictly between them. (8) Prove by induction that for any natural number n, n + 3