Let a and b be positive integers, and k an integer with O k<ab. Provide a...
Let m be a positive integer and let a and b be integers relatively prime to m with (ord m a , ord m b) )=1. Prove that ord m (ab)= (ord m a) (ord m b) (Hint: Let k=ord m(a),l=ord m(b), and n=ord m(ab). Then 1≡(ab)^kn≡b^kn mod m. What does this imply about l in relation to kn?
Let n be a positive integer. For each possible pair i, j of integers with 1 sisi<n, find an n x n matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.
Let n be a positive integer. For each possible pair i, j of integers with 1<i<i <n, find an n xn matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.
1) Let n and m be positive integers. Prove: If nm is not divisible by an integer k, then neither n norm is divisible by k. Prove by proving the contrapositive of the statement. Contrapositive of the statement:_ Proof: Direct proof of the contrapositive
Ok = (6) Let n be a positive integer. For every integer k, define the 2 x 2 matrix cos(27k/n) - sin(2nk/n) sin(2tk/n) cos(27 k/n) (a) Prove that go = I, that ok + oe for 0 < k < l< n - 1, and that Ok = Okun for all integers k. (b) Let o = 01. Prove that ok ok for all integers k. (c) Prove that {1,0,0%,...,ON-1} is a finite abelian group of order n.
Problem 3 Let n and k > l be positive integers. How many different integer solutions are there to x1 +...+ In = k, with all xi <l?
5. (a) Show that 26 = 1 mod 9. (b) Let m be a positive integer, and let m = 6q+r where q and r are integers with 0 <r < 6. Use (a) and rules of exponents to show that 2" = 2 mod 9 (c) Use (b) to find an s in {0,1,...,8} with 21024 = s mod 9.
Provide a bijective proof for the following identity: k()n m-1
What is wrong with the following proof that every positive integer equals the nex larger positive integer? "Proof," Let P(n) be the proposition that n = n + 1, Assume that P(k) is true, so that k = k + 1 . Add 1 to both sides of this equation to obtain k + 1-k + 2 . Since this is the statement P(k 1), It follows that P(n) is true for all positive integers n.
QUESTION C. (a) Let k be a field and let n be a positive integer. Define what is meant by a monomial ideal in k[x,...,zn]. 2. (b) State what it means for a ring R to be Noetherian. (c) State Hilbert's basis theorem. Give a proof of Hilbert's basis theorem using the fact if k is a field the polynomial ring kli,..., In] is Noetherian. 1S (a) Let k be a field and let n be a positive integer. Define...