g(1) geometric multiplicity of 1
a(1) algebraic multiplicity of 1
If these are not then please mention
Let n be a positive integer. For each possible pair i, j of integers with 1...
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.
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?
20 -{24R/1<x<1+ }}-(1,1+ 4) for each positive integer S: = ? a. US = ? i=1 4 b. O S = ? 11 c. Are S1, S2, S3, ... mutually disjoint? Explain. d. ÜS; = ? =1 72 e. n S = ? 1-1 00 f. US = ? 00 g. S; = ?
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.
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Let An = {j EN:j<n} for n = 1,2,.... What are D. and A.
Problem 1 (**) Find a positive integer m such that φ(m) < 0.1m.
I randomly pick two integers from 1 to n without replacement (n a positive integer). Let X be the maximum of the two numbers. (a) Find the probability mass function of X. (b) Find E(X) and simplify as much as possible (use formulas for the sum and sum of squares of the first n integers which you can find online).
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...