The contradiction method used to prove that the square root of a prime is irrational The contradiction method given in certain books to prove that sqare root of a prime is irrational also shows that sqare root of $4$ is irrational, so how is it acceptable?
e.g. Suppose $\sqrt{4}$ is rational,
$$\begin{align}
       \sqrt{4} &=p/q        \qquad\text{where pand q are coprimes} \\
        4 &=p^2/q^2\\
      4q^2&=p^2       \tag{1} \\
         4&\mid p^2\\
         4&\mid p\\
     \text {let }p&=4m  \qquad\text{for some natural no. m}  \\
       p^2&=16m^2\\
      4q^2&=16m^2   \qquad\text{(from (1) )}\\
       q^2&=4m^2\\
         4& \mid q^2\\
         4&\mid q
\end{align}
$$
but this contradicts our assumption that $p$ and $q$ are coprime since they have a common factor $p$. Hence  $\sqrt{4}$ is not rational. But we know that it is a rational. Why?
 A: If $4q^2 = p^2$, $4$ is a factor of $p^2$, but it does not follow that $4$ is a factor of $p$ only that $2$ is a factor of $p$. For example, $4$ is a factor of $36$, but $4$ is not a factor of $6$ (but $2$ is). 
In general, if $a$ is prime and $a$ is a factor of $pq$ then $a$ must be a factor of $p$ or a factor of $q$ (or both); in particular, if $a$ is a factor of $p^2$, then $a$ must be a factor of $p$. It then follows that if $m$ is a product of distinct primes, then if $m$ is a factor of $p^2$, $m$ must be a factor of $p$. So the method used to show that the square root of a prime number is irrational extends to numbers which are the product of distinct primes, but no further.
A: The step from $4\mid p^2$ to $4\mid p$ is wrong.  For example, take $p=6$.  Then $4\mid 6^2$ but it is not true that $4\mid 6$.
In general, $q\mid p^2$ implies $q\mid p$ only for squarefree $q$.  A number $q$ is "squarefree" if it is not divisible by any square larger than 1.
A: Let $p$ be prime and suppose that $\sqrt{p}$ is rational. Then $\sqrt{p}=\frac{r}{s}$ for some integers $r$ and 
$s$ with $s$ not zero. Then $s^2 p=r^2=q_1 \ldots q_n$ for some primes $q_i$ by the fundamental theorem of 
arithmetic. Note that $r^2$ and $s^2$ have an even number of prime factors by expanding squares. But then 
$s^2 p$ has an odd number of prime factors. This is a contradiction. Then $\sqrt{p}$ is irrational.
