p-Z-given-u-14p-3-v-5p-1-E-87x-31y-2-we-have-also-the-line-D-87x-31y-2-1-show-that-u-and-v-are-primes-between-them-i-mean-the-don-t-have-any-common-divisor-excepted-1-and-1-2-dedu

Question Number 122647 by mathocean1 last updated on 18/Nov/20

p \in Z . g i v e n :

u = 14 p + 3; v = 5 p + 1,

(E) : 87 x + 31 y = 2; w e h a v e

a l s o t h e l i n e (D) : 87 x - 31 y = 2

1) s h o w t h a t u a n d v a r e p r i m e s

b e t w e e n t h e m . (i m e a n t h e

{d o n}^{'} t h a v e a n y c o m m o n d i v i s o r

e x c e p t e d 1 a n d - 1 .)

2) d e d u c t t h a t 87 a n d 31 a r e

p r i m e s b e t w e e n t h e m .

3) S o l v e (E) .

4) D e t e r m i n a t e p o i n t s (x; y) \in (D)

w h i c h t h a t t h e i r c o r d o n n a t e s

x; y \in N a n d x ⩽ 100