In the period of finals ( and studying for finals ) yahoo answers have been flooded with homework and assignment problems 99.9% involve mundane “plug it into a formula” type of solutions. As such, these are of no particular interest to me or for this series. Hopefully, now that most students swapped book for beaches, there will be more interesting questions posted. Meanwhile, here are two short problems I picked some time ago

**Problem**: Prove that if is relatively prime to and ,then

**Solution**: Using it follows

Since is relatively prime to we know that

and it must be true that

Thus .

**Problem**: If and prove .

**Solution**: Since we know that for some . Similarly, for some . Notice, that both and must be odd. If, lets say, were even, then for some . But then and so .

Using and we have . But since both and are odd, their sum is even, i.e.

for some

It now follows that

and so