The document outlines a task to select three mathematical theories and provide proof for each using different proof techniques: direct, contrapositive, or contradiction. The theories include statements about even and odd integers related to multiplication and addition. It requires justification for the chosen proof methods for each statement.

1. Q1: Choose only three of the following theories and provide proof.
Do not use the same proof approach more than once (use direct, contrapositive, or
contradiction).
Justify your choice of proof approach for each theory.
A. If n is an even integer, then 3n 1 is an odd integer.
B. If 5x 7 is odd, then 9x + 2 is even
C. 11x 7 is even if and only if x is odd.