Purchase Solution

Congruences : Primes, Inverse Modulo, GCD and Wilson's theorem

Not what you're looking for?

Ask Custom Question

Please assist me with the attached congruence problems (hint: use Wilson's Theorem)

a)Prove if a,b,c Z, N and gcd (c, ) = , then ac bc(mod ) if and only if
a b (mod ).
b) Let a Z, N, and p > 2 be a prime. Prove that a is its own inverse modulo p if and only if a 1 (mod p ).

C) Let a,b Z, N .prove that ax b(mod ) has a solution if and only if gcd (a, ) b.

d) Suppose that 3(mod 4) is prime. Prove that

)! 1 (mod )
Hint( Use Wilson's Theorem)

Please see the attached file for the fully formatted problems.

Attachments
Purchase this Solution

Solution Summary

Congruences, Primes, Inverse Modulo, GCD and Wilson's theorem are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

Solution Preview

a)Prove if a,b,c elements of Z, n element of N and gcd (c, n) = g, then ac is congruent to bc(mod n ) if and only if
a is congruent to b (mod n/g ).
_____________________________________________

(i) if: suppose a is congruent to b (mod n/g). (note that g must divide n because g = gcd (c,n) so that n/g is an integer)
Then a = b + kn/g for some integer k (definition of congruence)
So ac = bc + kcn/g (multiplication, integers so we can commute and associate)
Since g = gcd (c,n), g divides both c and n; c/g is an integer; let c/n = j
Then ac = bc + kjn where kj is an integer, and thus ac is congruent to bc modulo n by definiton

(ii) only if: Suppose ac is congruent to bc modulo n
Then ac = bc + hn for some integer h, by definition of congruence
Since g = gcd (c,n), c = gd and n = gf for some integers f and d, and gcd (d,f) = 1
(by definition of gcd); note f = n/g as needed
Substituting, agd = bgd + hgf
Dividing out the g (g is an integer not equal to zero, because a gcd is always 1 or greater)
ad = bd + hf
Dividing out the d,
a = b + (hf)/d
Since a and b are integers, which are closed under additin and subtraction, (hf)/d must be an integer, d must divide hf.
Given that gcd (f,d) = 1 from above, d must divide h and h/d must be an integer
so ...

Purchase this Solution


Free BrainMass Quizzes
Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Probability Quiz

Some questions on probability