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parthenos

Member

Registered: 2008-08-25

Posts: 9

Abstract Algebra - integers modulo n (3 problems)

1.  Prove that if a = an10n + an-110n-1 + …+ a110 + a0 is any positive integer then
     a ≡ an + an-1 +…+ a1 + a0 (mod 9).  [Note: 10 ≡ 1 (mod 9)]

2.  Prove that the equation a2 + b2 = 3c2 has no solutions in nonzero integers a, b, and c.
   
    [Consider the equation mod 4 and show that a, b and c would all have to be divisible by 2.  Then each of a2, b2, c2 has a factor of   
        4 and by dividing through by 4 show that there would be a smaller set of solutions to the original equation.  Iterate to reach a
        contradiction]

3.  Let n є Z, n > 1, and let a є Z with 1 ≤ a ≤ n.  Prove if a and n are not relatively prime, there exists an integer b with 1 ≤ b < n such that ab ≡ 0 (mod n) and deduce that there cannot be an integer c such that ac ≡ 1(mod n).

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Abstract Algebra - integers modulo n (3 problems)

#1
Note that

.

for some integer K.

#3
Let p be a prime dividing both a and n and set

. Then

and

.

Last edited by JaneFairfax (2008-08-27 00:43:28)