How many possible values of $(p + q + r)$ are there satisfying $\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r} =…
How many possible values of $(p + q + r)$ are there satisfying $\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r} = 1$, where $p$, $q$ and $r$ are natural numbers (not necessarily distinct)?
None
One
Three
More than three
Solution
The natural-number solutions of $\dfrac{1}{p}+\dfrac{1}{q}+\dfrac{1}{r}=1$ (unordered) are: $(2,3,6)$ giving sum 11, $(2,4,4)$ giving sum 10, and $(3,3,3)$ giving sum 9. These yield three distinct values of $(p+q+r)$: 9, 10 and 11. Hence the answer is three.