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)deduct that 87 and 31 are primes between them. 3)Solve (E). 4)Determinate points (x;y)∈ (D) which that their cordonnates x;y ∈ N and x≤100


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

$p \in Z . g i v e n :$ $u = 14 p + 3; v = 5 p + 1,$ $(E) : 87 x + 31 y = 2; w e h a v e$ $a l s o t h e l i n e (D) : 87 x - 31 y = 2$ $1) s h o w t h a t u a n d v a r e p r i m e s$ $b e t w e e n t h e m . (i m e a n t h e$ ${d o n}^{'} t h a v e a n y c o m m o n d i v i s o r$ $e x c e p t e d 1 a n d - 1 .)$ $2) d e d u c t t h a t 87 a n d 31 a r e$ $p r i m e s b e t w e e n t h e m .$ $3) S o l v e (E) .$ $4) D e t e r m i n a t e p o i n t s (x; y) \in (D)$ $w h i c h t h a t t h e i r c o r d o n n a t e s$ $x; y \in N a n d x ⩽ 100$
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