$\dfrac{pq}{r}+\dfrac{qr}{p}+\dfrac{pr}{q}\in\Bbb Z\,\Rightarrow\, \dfrac{pq}{r},\dfrac{qr}{p},\dfrac{pr}{q}\in\Bbb Z$
Hint $ $ The hypothesis implies $\,\large {\big(x-\frac{pq}r\big)\big(x-\frac{qr}p\big)\big(x-\frac{pr}q\big)}\,$ has all integer coefficients, therefore the Rational Root Test implies that its rational roots are integers.