please do part a) number 1 Problem 1. (a) Prove that every prime p greater than

Question

please do part a) number 1

Problem 1. (a) Prove that every prime p greater than 2 satisfies p mod 8 -r where r is 1, 3, 5, or 7! Thus primes 2 may be divided into four classes depending on their remainders when divided by 8. No Maple calculations are necessary here but write out your proof using Maple as a word processor. Hint: Can p 2 be a prime if p mod 8 r when r 0, 2, 4 or 6? Note that p mod 8 r means that p-8q + r where r = 0, 1, 2, 3, 4, 5, 6, or 7.1 (b) Set N:-100. (i) Form four lists L[1], L[3], L[5], LI7] where L[r] contains all primes 2 p

Explanation / Answer

RTP:-any prime no greater than 2 satisfy p mod 8=r where r=1,3,5 or 7

solution:- p mod 8=r means that p=8q+r .....(1) where r=0,1,2,3,4,5,6, or 7

now as we know by euclidean division algorithm , if a=bq+r then gcd(a,b)=gcd(b,r)

similarly gcd(p,8) and gcd(8,r) should be the same in equ(1)

since p is a prime number then gcd(p,8)=1

so the value of r will be the values with whom gcd(8,r) will be 1.

hence r can be 1,3,5 or 7 as gcd(8,1)=gcd(8,3)=gcd(8,5)=gcd(8,7)=1

hence proved.

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