number theory - Prove that $sqrt{5n+2}$ is irrational

I'm trying to follow this answer to prove that $\sqrt{5n+2}$ is irrational. So far I understand that the whole proof relies on being able to prove that $(5n+2)|x^2 \implies (5n+2)|x$ (which is why $\sqrt{4}$ doesn't fit, but $\sqrt{7}$ etc. does), this is where I got stuck. Maybe I'm overcomplicating it, so if you have a simpler approach, I'd like to know about it. :)

A related problem I'm trying to wrap my head around is: Prove that $\frac{5n+7}{3n+4}$ is irreducible, i.e. $(5n+7)\wedge(3n+4) = 1$.

Answer

Well, one way to say it is irrational is to see that $5n+2$ isn't an integer square for any $n\in\mathbb{Z}$ (it only finish in $2$ or $7$). The other way you're trying lets you the same ending $(p,q\in \mathbb{Z}, q\neq 0)$:
\begin{align*}
\sqrt{5n+2}=\frac{p}{q}&& \\
5n+2=\frac{p^2}{q^2} &&(1)\\
q^2(5n+2)=p^2 && (2)

\end{align*}

Let
$$p=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_t^{\alpha_t}$$ $$q=q_1^{\beta_1}q_2^{\beta_2}\dots q_s^{\beta_s}$$

where $p_i,q_j$ are primes and $\alpha_i,\beta_j$ are positive integers.

From $(1)$, $p^2/q^2$ is an integer (is equal to $5n+2$), so you have that the $q_i$ are certain primes $p_j$. Renaming the prime factors in a way such that $q_i=p_i$, you can let you the fraction (considering that $t>s$)

$$\frac{p^2}{q^2}=\frac{p_1^{2\alpha_1}p_2^{2\alpha_2}\dots p_t^{2\alpha_t}}{q_1^{2\beta_1}q_2^{2\beta_2}\dots q_s^{2\beta_s}}=p_1^{2(\alpha_1-\beta_1)}p_2^{2(\alpha_2-\beta_2)}\dots p_t^{2(\alpha_s-\beta_s)}\dots p_t^{2\alpha_t}=5n+2$$

then this implies that $5n+2$ is a square, but by the original statement, it can't be.