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    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...
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    Number Theory - Affine Cipher

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

    Homework Statement Below is an example of ciphertext obtained from an Affine Cipher. Determine the plaintext. KQEREJEBCPPCJCRKIEACUZBKRVPKRBCIBQCARBJCVFCUPKRIOFKPACUZQEPBKRXPEIIEABDKPBCPFCDCCAFIEABDKPBCPFEQPKAZ BKRHAIBKAPCCIBURCCDKDCCJCIDFUIXPAFFERBICZDFKABICBB...
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    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...
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    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)...
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    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 +...
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    Adjacency Matrix Proof

    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...
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    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...
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    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...
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    Steiner Triple System

    Homework Statement Use the doubling construction to construct a Steiner Triple System of order 19. (Exhibit the blocks.) The Attempt at a Solution My first question is, what is a doubling construction? I can't find it in my book or on the internet.
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    Recurrence Relation

    Homework Statement The sequence f_n is defined by f_0=1, f_1=2 and f_n=-2f_{n-1}+15f_{n-2} when n \geq 2. Let F(x)= \sum_{n \geq 2}f_nx^n be the generating function for the sequence f_0,f_1,...,f_n,... Find polynomials P(x) and Q(x) such that F(x)=\frac{P(x)}{Q(x)} The Attempt at a...
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    Recurrence Relation to Non-recursive Formula

    Homework Statement The sequence f_n is defined by f_0=f_1=2 and f_n = (\frac{f_{n-1}+2f_{n-2}}{6}), when n\geq2. Find a non-recursive formula for f_n The Attempt at a Solution Well I have solved for the closed formula of the generating function which I will call g(x) so...
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    Inclusion/Exclusion Combinatorics

    Homework Statement Determine the number of permutations of {1,2,3,4,5,6,7} in which exactly four integers are in there natural positions. The Attempt at a Solution Would this be solved by using the Inclusion/Exclusion Principle and finding \left|S\right| - \sum \left|A_{1}\right|...
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    Combinatorics Proof

    Homework Statement Prove that n5^n = \frac{5}{4} \sum_{k=0}^{n}k\begin{pmatrix}n\\k\end{pmatrix}4^k (Hint: First Expand (1+x^2)^n) The Attempt at a Solution So if I expand that I get (1+x^2)^n = (1+x^2)(1+x^2)...(1+x^2) n times so it equals \sum_{k=0}^n (1+x^2) Not sure where to...
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    Basic Combinatorics Inclusion-Exclusion Principle Clarification

    Homework Statement How many integers between 1 and 2009, inclusive are (a) not divisible by 3,2, and 10 (b) not divisible by 3,2, or 10? Homework Equations The number of objects of the set S that have none of the properties is given by the alternating expression: \mid S \mid -\sum...
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    Combinatorics-binomial coefficients

    Homework Statement Find integers a,b, and c such that m^{3} = a*nCr(m,3)+b*nCr(m,2)+c*nCr(m,1) for all m. Then sum the series 1^3+2^3+3^3+...+n^3 Homework Equations I think I need to use m^{3} = a*nCr(m,3)+b*nCr(m,2)+c*nCr(m,1) and...
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    Combinatorics-binomial expansion?

    Homework Statement Let A(x) = \sum_{k>=0}a_{k}x^k and B(x) = \sum_{k>=0}b_{k}x^k show that: A(X)B(X) = \sum_{k>=0}(\sum_{i=0}a_{i}b_{k-i})x^k The second sum sign in the answer should be from i=0 to k. The Attempt at a Solution I factored out like terms and then...
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