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Let $p,q,r$ be integers such that the symmetric sum of fractions $\dfrac{pq}{r}+\dfrac{qr}{p}+\dfrac{pr}{q}$ is an integer.

Prove that each of the numbers: $\dfrac{pq}{r},\dfrac{qr}{p},\dfrac{pr}{q}$ is an integer.

How to do this?

Anne Bauval
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mathlover
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1 Answers1

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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.

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Bill Dubuque
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  • is " Rational Root Test implies that its rational roots are integers" true only when leading coefficient is $1$, and which is with our question ($x^3$)? – mathlover Jan 25 '17 at 12:31
  • @Ayush Yes, if the leading coefficient is $,c,$ then RRT implies that the denominator $,d,$ of a least-terms rational root must divide $c$. Only the monic case $,c = \pm1.,$ has a unique positive divisor $(= 1),, $ since all positive divisors $d$ of $c$ can occur as a denominator, e.g. $,(dx-1)(ex^{n-1}-1),$ has root $,x = 1/d.\ $ – Bill Dubuque Jan 25 '17 at 15:11
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    @Num If $,x^3+bx^2+cx+d,$ has all integer coefficients then every rational root is an integer, by RRT.. – Bill Dubuque Jan 25 '17 at 16:02
  • @Num All the rational roots are integral (this is vacuously true.when it has no rational roots). – Bill Dubuque Jan 25 '17 at 16:07