If every pair from among the equations $ x^{2}+px+qr=0 $, $ x^{2}+qx+rp=0 $, and $ x^{2}+rx+pq=0 $ has a common root, then the product of three common roots is

Let the common root of the first two equations be $\alpha$, the common root of the last two be $\beta$, and the common root of the first and third be $\gamma$. Subtracting the equations in pairs gives $(p-q)x = r(p-q)$, which means $\alpha = r$. Similarly, subtracting other pairs yields $\beta = p$ and $\gamma = q$. The product of the three common roots is $\alpha\beta\gamma = rpq = pqr$.

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