Since the OP has mentioned being new to proof writing, I'll try to help him/her get started by providing a example.
It looks like you(the OP) are having difficulty moving from examples you have tried ( assuming you have done an extensive trial ) I'm going to help you get started a generalization of ideas.
Dennis Gulko gave a starting point. After reading what's written here, go back to his comment and think about it.
Ask yourself: what do I have and what is asked for.
As data ( hypotheses ) you have $a$ and $b$ being odd.
Now the question is find out whether $\frac{a}{b} = c$ is either odd or even.
To conjecture means you try as many instances of the problem as you can then formulate a conjecture ( a statement you assume to be true but you haven't proved yet ).
Example of a conjecture is If a is an odd integer and b is an odd integer then a+b is even.
We don't know for sure but $3+3$ is even and we can try with others. ( In your case, do multiple trials. )
You see, the value of the conjecture is that it allows you to have something to prove. A sentence like If a is an odd integer and b is an odd integer then a+b is even or odd.
is not a useful conjecture because it is always true. There is no third possibility.
After you have a conjecture it helps to know the definition of key concepts involved. E.g what is an odd number, an even number? The good news in mathematics is that definition are written not to be ambiguous.
In the conjecture i provided, we need to know exactly what an odd and an even numbers are.
And we say: let a=2n+1 and b=2m+1 be two odd numbers. We have to prove that a+b is equal to a third number c=2k which is even.
Then we have:
We have a=2n+1 and b=2m+1, then c=a+b=2n+1+2m+1=2(n+m)+2=2(n+m+1)=2k. qed
That is a flow you might use to get started proving things.