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Problem 2: Indirect and Euclidean proofs (40 pts) For the following problems, you must use an...

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Advance Math

Problem 2: Indirect and Euclidean proofs (40 pts) For thefollowing problems, you must use an indirect proof technique.

(a) (10 pts) Prove indirectly that, if a 2 is a multiple of 31,then so is a. Your proof should not consist of 30 cases – thisincludes absolutely no implied cases using horizontal dots (· · ·)and/or vertical dots (. . .).

(b) (15 pts) Using the result of question (a), prove that ? 31is not a rational Q using the Euclidean method.

(c) (15 pts) Using the result of question (a), prove that ? 31is not a rational Q using the Unique Prime FactorizationTheorem.

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