For-p-q-and-r-prime-numbers-satisfying-p-q-1-r-1-1064-r-p-1-q-1-1554-find-the-value-p-q-1-r-

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

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

Answered by A5T last updated on 06/Nov/24

p∣1064=2^3 ×7×19; p+1∣1554=2×3×7×37 ⇒p=2⇒(q+1)(r+1)=532=4×7×19 ⇒r(q+1)=2×7×37⇒r=37⇒q=13 ⇒p(q+1)r=2×14×37=1036

$${p}\mid\mathrm{1064}=\mathrm{2}^{\mathrm{3}} ×\mathrm{7}×\mathrm{19};\:{p}+\mathrm{1}\mid\mathrm{1554}=\mathrm{2}×\mathrm{3}×\mathrm{7}×\mathrm{37} \\ $$$$\Rightarrow{p}=\mathrm{2}\Rightarrow\left({q}+\mathrm{1}\right)\left({r}+\mathrm{1}\right)=\mathrm{532}=\mathrm{4}×\mathrm{7}×\mathrm{19} \\ $$$$\Rightarrow{r}\left({q}+\mathrm{1}\right)=\mathrm{2}×\mathrm{7}×\mathrm{37}\Rightarrow{r}=\mathrm{37}\Rightarrow{q}=\mathrm{13} \\ $$$$\Rightarrow{p}\left({q}+\mathrm{1}\right){r}=\mathrm{2}×\mathrm{14}×\mathrm{37}=\mathrm{1036} \\ $$

Facebook Tweet Pin

For-p-q-and-r-prime-numbers-satisfying-p-q-1-r-1-1064-r-p-1-q-1-1554-find-the-value-p-q-1-r-

Leave a Reply Cancel reply