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Question Number 213463 by golsendro last updated on 06/Nov/24

$$\mathrm{For}\mathrm{p},\mathrm{q}\mathrm{and}\mathrm{r}\mathrm{prime}\mathrm{numbers}$$$$\mathrm{satisfying}\{\begin{array}{l}\mathrm{p}(\mathrm{q}+1)(\mathrm{r}+1)=1064\\ \mathrm{r}(\mathrm{p}+1)(\mathrm{q}+1)=1554\end{array}$$$$\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{p}(\mathrm{q}+1)\mathrm{r}$$

Answered by A5T last updated on 06/Nov/24

$$p\mid 1064=2^3\times 7\times 19;p+1\mid 1554=2\times 3\times 7\times 37$$$$\Rightarrow p=2\Rightarrow (q+1)(r+1)=532=4\times 7\times 19$$$$\Rightarrow r(q+1)=2\times 7\times 37\Rightarrow r=37\Rightarrow q=13$$$$\Rightarrow p(q+1)r=2\times 14\times 37=1036$$

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