Exercise 2

For each of the statements below, say what method of proof you should use to prove them. Then say how the proof starts and how it ends. Bonus points for filling in the middle.

1. There are no integers $x$ and $y$ such that $x$ is a prime greater than 5 and $x = 6y + 3$.

2. For all integers $n$, if $n$ is a multiple of 3, then $n$ can be written as the sum of consecutive integers.

3. For all integers $a$ and $b$, if $a^2 + b^2$ is odd, then $a$ or $b$ is odd.