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Problem 3. $[30$ points] Recall that a rational number can be put in the form $qp?$ where $p$ and $q$ are integers and $q?=0$. Prove the following for any rational number, $x$ : a. [10 points] If $x$ is rational, then $x?5$ is rational b. [ 10 points] If $x?5$ is rational, then $x/3$ is rational c. [ 10 points] If $x/3$ is rational, then $x$ is rational Problem 4. [20 points] Consider the statement: For all integers $m$ and $n$, if $m?n$ is odd, then $m$ is odd or $n$ is odd. a. [10 points] Prove the statement using a proof by contrapositive b. [10 points] Prove the statement using a proof by contradiction

Problem 2We have to prove that if a is odd integer then 3a+1 is e