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Do there exist positive integers $x,y,z$ such that $x,y$ have exactly $1000$ common positive divisors, $y,z$ have exactly $720$ common positive divisors, and $z,x$ have exactly $350$ common positive divisors?

If a number has a prime factorization $p_1^{a_1}\dots p_n^{a_n}$, then the number of divisors is $(a_1+1)\dots(a_n+1)$. So if $\gcd(x,y)$ is written in this form, then $p_1^{a_1}\dots p_n^{a_n}=1000$. Similarly for $\gcd(y,z)$ and $\gcd(z,x)$.

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Let $a,b,c$ be distinct primes (e.g., $a = 2$, $b = 3$, $c = 5$), and let

\begin{align*} x &= a^{999}b^{349}\\[6pt] y &= a^{999}c^{719}\\[6pt] z &= b^{349}c^{719}\\ \end{align*}

Then

  • $\;\text{gcd}(x,y)=a^{999}$, so $x,y$ have $1000$ common positive integer divisors.
  • $\;\text{gcd}(y,z)=c^{719}$, so $y,z$ have $720$ common positive integer divisors.
  • $\;\text{gcd}(z,x)=b^{349}$, so $z,x$ have $350$ common positive integer divisors.
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