[SOLVED] Proving a theorem without using contradiction

**racewithferrari** · November 16th 2009, 03:30 PM

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**Plato** · November 16th 2009, 03:41 PM

Originally Posted by racewithferrari

The following theorem has the form of an implication.
If n is an integer and 3n+2 is even, then n is even.

Suppose that $3n+2=2k$ then $3n=2k-2$ .
What does that tell you?

**Drexel28** · November 16th 2009, 03:42 PM

Originally Posted by racewithferrari

The following theorem has the form of an implication.

If n is an integer and 3n+2 is even, then n is even.

Give a direct proof of this theorem without using contradiction.

Since $3n+2$ is even we may state WLOG that $3n+2=2z\implies 3n=2(z-1)$ and since $2\nmid 3$ it must be true that $2\mid n$ . The conclusion follows.

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