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Show all work so that I can learn from this and understand. Please show how you got the correct answer, I will be checking this so I need to be able to follow along. I will rate low or not at all if you give me an ambiguous answer, this includes not being able to solve it at all if you just assume I know something you don't.

Use proof by contradiction to show the following: If a is an arbitrary integer and 3a+7 is even, then a is odd. Use direct proof to show that if a is odd, then 3a+7 is even. Given the results from question a and this question, are the two following statements logically equivalent: "a is an odd integer" and "a is an integer and 3a+7 is even an even integer"?

Explanation / Answer

a)lets take a as even

a=2n,3a+7=3(2*n)+7=6*n+7=odd which is false

so a is odd

b)lets take a as odd=2*n+1

3*a+7=3(2*n+1)+7=6*n+10 which is even

both statements are logically equivalent, 2nd equation implies a is an odd integer

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