Q2 Let $p$ be a prime number, and let $x$ and $y$ be positive integers. Then we are to find all triples of $p$, $x$ and $y$ which satisfy $$\frac{p}{x} + \frac{7}{y} = p.$$ We can transform this equation into $$(x - \mathbf{N})(py - \mathbf{OO}) = \mathbf{OP}.$$ From this, we obtain $$x - \mathbf{N} = \mathbf{Q} \text{ or } \mathbf{R}, \quad (\text{note: have } \mathbf{Q} < \mathbf{R})$$ and hence $$x = \mathbf{S} \text{ or } \mathbf{T}. \quad (\text{note: have } S < T)$$ First, if $x = S$, then $$p = \mathbf{U}, \quad y = \mathbf{V}$$ or $$p = \mathbf{W}, \quad y = \mathbf{X}. \quad (\text{note: have } \mathbf{U} < \mathbf{W})$$ Next, if $x = T$, then $$p = \mathbf{Y}, \quad y = \mathbf{Z}.$$
Q2 Let $p$ be a prime number, and let $x$ and $y$ be positive integers. Then we are to find all triples of $p$, $x$ and $y$ which satisfy
$$\frac{p}{x} + \frac{7}{y} = p.$$
We can transform this equation into
$$(x - \mathbf{N})(py - \mathbf{OO}) = \mathbf{OP}.$$
From this, we obtain
$$x - \mathbf{N} = \mathbf{Q} \text{ or } \mathbf{R}, \quad (\text{note: have } \mathbf{Q} < \mathbf{R})$$
and hence
$$x = \mathbf{S} \text{ or } \mathbf{T}. \quad (\text{note: have } S < T)$$
First, if $x = S$, then
$$p = \mathbf{U}, \quad y = \mathbf{V}$$
or
$$p = \mathbf{W}, \quad y = \mathbf{X}. \quad (\text{note: have } \mathbf{U} < \mathbf{W})$$
Next, if $x = T$, then
$$p = \mathbf{Y}, \quad y = \mathbf{Z}.$$