# Search results

1. ### Number Theory

Yup, I don't know why I didn't see that right away. Thanks!
2. ### Number Theory

Homework Statement The idea of this problem is to investigate the solutions to x^2=1 (mod pq), where p,q are distinct odd primes. (a) Show that if p is an odd prime, then there are exactly two solutions (mod p) to x^2=1 (mod p). (Hint: difference of two squares) (b) Find all pairs...
3. ### Number Theory - Affine Cipher

Nevermind, I figured it out.
4. ### Number Theory - Affine Cipher

Homework Statement Decipher the following text KQEREJEBCPPCJCRKIEACUZBKRVPKRBCIBQCARBJCVFCUPKRIOF KPACUZQEPBKRXPEIIEABDKPBCPFCDCCAFIEABDKPBCPFEQPKAZ BKRHAIBKAPCCIBURCCDKDCCJCIDFUIXPAFFERBICZDFKABICBB ENEFCUPJCVKABPCYDCCDPKBCOCPERKIVKSCPICBRKIJPKABI Homework Equations I know that...
5. ### Affine Cipher

Homework Statement Below is an example of ciphertext obtained from an Affine Cipher. Determine the plaintext. KQEREJEBCPPCJCRKIEACUZBKRVPKRBCIBQCARBJCVFCUPKRIOFKPACUZQEPBKRXPEIIEABDKPBCPFCDCCAFIEABDKPBCPFEQPKAZ BKRHAIBKAPCCIBURCCDKDCCJCIDFUIXPAFFERBICZDFKABICBB...
6. ### Number Theory - Repunits

I figured out that for a its if and only if the number of digits is a multiple of d, where d divides b-1. For b, would it just be if and only if the alternating sum comes out to a multiple of d, where d divides b+1?
7. ### Number Theory - Repunits

Well for b=1 then \alpha will be positive so \beta will have to be the negative of \alpha The opposite is true for b=-1, \alpha will be negative and \beta is going to be the positive of that number...
8. ### Number Theory - Repunits

We haven't covered the remainder theorem yet...
9. ### Number Theory - Repunits

Homework Statement A base b repunit is an integer with base b expansion containing all 1's. a) Determine which base b repunits are divisible by factors b-1 b) Determine which base b repunits are divisible by factors b+1 Homework Equations R_{n}=\frac{b^{n}-1}{b-1} The Attempt...
10. ### Number Theory Questions

For 3 part a, additive order of a modulo n is defined to be the smallest positive integer m that satisfies the congruence equation m*a \cong 0 (mod n). So in this case it'd be better to write a modulo n as m*a \cong 0 (mod n). m would be our additive order which means since n=78 our m=78/a?
11. ### Number Theory Questions

I am not really sure what you are getting at.
12. ### Number Theory Questions

that ax gives us a number equal to the equivalence class , i.e. divisible by 78
13. ### Number Theory Questions

That is the problem, I haven't been able to find one in the book or in my notes.
14. ### Number Theory Questions

So for question 1.  can occur because 2^{2}+2^{2}= 0 mod 4.  can occur because 2^{2}+1^{2}= 1 mod 4. Is  the only one that can not occur? As for question 2a, I went through and squared all numbers from 1 to 20, the only options...
15. ### Number Theory Questions

Homework Statement 1) What are the possible values of m^{2} + n^{2} modulo 4? 2) Let d_{1}(n) denote the last digit of n (the units digit) a) What are the possible values of d_{1}(n^{2})? b) If d_{1}(n^{2})=d_{1}(m^{2}), how are d_{1}(n) and d_{1}(m) related? 3) a)...
16. ### Basic differential equations

Homework Statement Determine whether the given first-order differential equation is linear in the indicated dependent variable by matching it with the first differential equations given in #7. Homework Equations #7 (sin \theta)y^{'''} - (cos \theta)y^{'} = 2 #10 (u)dv +...

so A^3[i,j] would be the total number of ways to get from i to j transversing exactly 3 edges.

A^2 would be the total number of ways to get from i to j wouldn't it?

I am really not getting this, the matrix multiplication makes sense but I don't get how to interpret it.

A^2[i,j] = A[i,j]*A[i,j]. Wouldn't this double the number of ways to get from i to j?

I am not sure what index notation is but are you trying to get at the fact that each entry in the matrix will then be 2^n-1 if we are looking at A^n?

I was assuming I was working with a simple graph only.

Well the adjacency matrix is how many edges between the two vertices so A[i,j] would be 1

Homework Statement Let A be the adjacency matrix for the graph G=(V,E) (a) Show that A^3[i,i] equals twice the number of triangles containing vertex i. (A triangle is a cycle of length 3 (b) Find an interpretation for A^3[i,j] when i does not equal j similar to the above. Prove that...
26. ### Graph Theory Proof

Homework Statement Prove that a graph is a tree if and only if it has no cycles and the insertion of any new edge always creates exactly one cycle. The Attempt at a Solution Assume that a graph G is connected and contains no vertices with a degree of zero. So would I get my proof by...
27. ### Graph Theory

Ok, so after getting the diagram, if I use that theorem to show G is Eulerian it is as easy as saying since G is a connected graph where the 2-element vertices have a degree of 2 while the 3-element vertices have a degree of 4 so they all have an even number of vertices making it Eularian...
28. ### Graph Theory

Ok, thats what I figured but I how would I find |A and B|? Is there some kind of ordered pair subtraction I am forgetting about or something? For example if I take A = {1,2} and B = {1,3} how do I find what |A and B| is? Does the method also work if I was to replace B with a 3-element set?
29. ### Graph Theory

Homework Statement Let X = {1,2,3,4} and let G = (V,E) be the graph whose vertices are the 2-element and 3-element subsets of X and where A is adjacent to B if |A and B| = 2. That is: V = nCr(X,2) or nCr(X,3) E = {{A,B}:A,B\inV and |A\capB|=2} (a) Draw the diagram of the graph G...
30. ### Steiner Triple System

Alright well I found out a doubling construction is just the fact that if an STS of order v, STS(v) exists then so does an STS(2v+1). So that means that my STS(19) is the same as a STS(39). Not sure what to do from there though...