$r$ is a number such that $r=p^a$. If the sum of some fractions equal to $1$ and one of the denominators is divisible by $r$ then there is another denominators that is exactly divisible $r$.
It seems to be really easy but I cannot prove it for example:
$\frac{7}{12}+\frac{4}{15}+\frac{3}{20}=1$
You can see here we have two denominators that are divisible by $4$ or two denominators that are divisible by $3$.Any hints?
Hint: suppose there is one, and only one, term with denominator divisible by $p^a$. Let $L$ be the least common multiple of all denominators. Multiply both sides by $L/p$. Then the RHS is an integer and the LHS is not.