$\large\frac{1}{p}+\frac{1}{q}+\frac{1}{pq}=\frac{1}{n}$

Over all natural numbers $n$ , how many ordered pairs of primes $(p,q)$ satisfy the equation above?

3
2
Infinitely Many
4
1
9

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$\frac{1}{p}+\frac{1}{q}+\frac{1}{pq}=\frac{1}{n}$

or, $\frac{p+q+1}{pq}=\frac{1}{n}$

or, $n(p+q+1)=pq$

As $p,q$ are primes so four cases are here

Case 1$n=1$ & $p+q+1=pq$

Case 2$n=p$ & $p+q+1=q$

Then $p=-1$

Which is not possible.

Case 3$n=pq$ & $p+q+1=1$

Then $p+q=0$

Which is not possible.

Case 4$n=q$ & $p+q+1=p$

Then $q=-1$

which is not possible.

So only

Case 1is possible.Taking it we will get $\boxed{2}$ ordered pairs $(2,3),(3,2)$