False Proof that $\sqrt{4}$ is Irrational

Everyone with any basic knowledge of number theory knows the classic proof of the irrationality of $\sqrt{2}$. Curious about generalizations using elementary methods, I looked up the irrationality of $\sqrt{3}$, and found the following:

Say $ \sqrt{3} $ is rational. Then $\sqrt{3}$ can be represented as $\frac{a}{b}$, where a and b have no common factors.

So $3 = \frac{a^2}{b^2}$ and $3b^2 = a^2$. Now $a^2$ must be divisible by $3$, but then so must $a $ (fundamental theorem of arithmetic). So we have $3b^2 = (3k)^2$ and $3b^2 = 9k^2$ or even $b^2 = 3k^2 $ and now we have a contradiction.

Such a proof follows the same basic logic as the proof for $\sqrt{2}$, except for using the fundamental theorem of arithmetic to replace and generalize the trivial fact that $n$ is even if $n^2$ is even. However, when I apply this proof format to $\sqrt{4} $ (which is clearly an integer and thus rational) I get the following:

Say $ \sqrt{4} $ is rational. Then $\sqrt{4}$ can be represented as $\frac{a}{b}$, where a and b have no common factors.

So $4 = \frac{a^2}{b^2}$ and $4b^2 = a^2$. Now $a^2$ must be divisible by $4$, but then so must $a $ (fundamental theorem of arithmetic). So we have $4b^2 = (4k)^2$ and $4b^2 = 16k^2$ or even $b^2 = 4k^2 $, which implies that $b=4n$ by the fundamental theorem. Now we have a contradiction (since can note that both $a$ and $b$ are divisible by $4$ and we assumed they were coprime)

This proof is clearly false, yet I fail to see where it differs. Where does it do so?

Just because $a^2$ is divisible by $4$, that doesn't mean $a$ is.

Your error is stating that if $a^2$ is divisible by 4 so must $a$ be. The fundimental theorem states if a prime $p $ divides $ab $ then $p $ must divide $a $ or $p$ must divide $b $. That is true because $p $ is indivisable.

But if $p $ is composite it doesn't hold. $p$ could equal $jk $ and $j$ could divide $a $ and $k $ divide $b $. Example: 3 divides 4 times 9 so 3 either divides 4 or 3 divides 9 because 3 is prime. But 6 divides 4 times 9 but 6 neither divides 4 nor 9 but instead 3 divides 9 while 2 divides 4 so 6=2 times 3 divide 4 times 9.

So 4 divides $a^2$ means $2*2$ divides $a*a$ so 2 divides $a $ is all you can conclude with certainty. ... because 4 is not prime.

Actually because 4 is not square free. All of its prime factors must divide into $a$ but the square powers can be distributed among (and are) distributed among the square powers of $a$.