Query about Reductio Ad Absurdum If we use the method of contradiction(i.e.Reductio Ad Absurdum), and if one of our assumptions is wrong, does that mean that all our assumptions are wrong and is the statement or hypothesis proved?
 A: You are correct: once we obtain a contradiction, we don't know which of our assumptions was false.  Any or all of them might be.  But the claim that $a$ and $b$ have no common factor is not an assumption.
In the proof that $\sqrt2$ is irrational, we start by supposing that $\sqrt2$ is rational.  A rational number is one that can be written as a fraction, so we have $\sqrt2 = \frac pq$  for some integers $p$ and $q$, and $q\ne 0$. 
Now what if $p$ and $q$ had a common factor?  Then we could take the fraction $$\sqrt2 = \frac pq$$ and cancel the common factor from the numerator and denominator and obtain an equal fraction $$\sqrt 2 = \frac ab$$ which was in lowest terms.  And then $a$ and $b$ would have no common factor greater than 1, because that is what it means to write a fraction in lowest terms.  Since every fraction can be written in lowest terms, we can do this no matter what $p$ and $q$ were.
So if $\sqrt 2$ is rational, we can find integers $a$ and $b$ with $\sqrt2=\frac ab$ and $a$ and $b$ have no common factor bigger than 1.  This is not an assumption; it's how fractions work.
And if we get nonsense later on, that they have common factors anyway, we know it must be nonsense.  It couldn't be because $a$ and $b$ had a common factor bigger than 1, because we know there must be some pair of integers  that don't have a common factor bigger than 1; we can obtain them by reducing the original fraction to lowest terms, and then call them $a$ and $b$.  It can't actually be that $a$ and $b$ really had a common factor all along, because if there had been one, we would already have canceled the factor and used the names $a$ and $b$ for the result instead.  So the problem must be something else.
A: I'm not sure I understand the question. However:
The only assumption made is that $p/q$ has no common factors, and then this is contradicted by showing that they must. 
This means that $\sqrt{2}$ cannot be written as a reduced fraction (i.e. with no common factors in $p$ and $q$). Since every fraction can be written in reduced form, there is no fraction equal to $\sqrt{2}$ - it does not help to switch to "another" fraction $a/b$.
