Question Number 122647 by mathocean1 last updated on 18/Nov/20
$${p}\:\in\:\mathbb{Z}.\:{given}: \\ $$$${u}=\mathrm{14}{p}+\mathrm{3}\:;\:{v}=\mathrm{5}{p}+\mathrm{1}, \\ $$$$\left({E}\right):\mathrm{87}{x}+\mathrm{31}{y}=\mathrm{2}\:;\:{we}\:{have}\: \\ $$$${also}\:{the}\:{line}\:\left({D}\right):\:\mathrm{87}{x}−\mathrm{31}{y}=\mathrm{2} \\ $$$$\left.\mathrm{1}\right){show}\:{that}\:{u}\:{and}\:{v}\:{are}\:{primes} \\ $$$${between}\:{them}.\left({i}\:{mean}\:{the}\:\right. \\ $$$${don}'{t}\:{have}\:{any}\:{common}\:{divisor}\: \\ $$$$\left.{excepted}\:\mathrm{1}\:{and}\:−\mathrm{1}.\right) \\ $$$$\left.\mathrm{2}\right){deduct}\:{that}\:\mathrm{87}\:{and}\:\mathrm{31}\:{are} \\ $$$${primes}\:{between}\:{them}. \\ $$$$\left.\mathrm{3}\right){Solve}\:\left({E}\right). \\ $$$$\left.\mathrm{4}\right){Determinate}\:{points}\:\left({x};{y}\right)\in\:\left({D}\right)\: \\ $$$${which}\:{that}\:{their}\:{cordonnates} \\ $$$${x};{y}\:\in\:\mathbb{N}\:{and}\:{x}\leqslant\mathrm{100} \\ $$$$ \\ $$