Math Is Fun Forum

Let σ be the m-cycle (1 2 …m).  Show that σi is also an m-cycle if and only if i is relatively prime to m.

I need some directions please.

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).

Can someone give me at least start on this?  I am not sure where to begin. 

I want to find the largest power of p (p is prime) which will divide n!

Thanks for your help

I need help solving the following problem.

For the following pair of integers a and n, show that a is relatively prime to n and determine the multiplicative inverse of ā in Z/nZ.

a = 69, n = 89