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...
Homework Statement
Decipher the following text
KQEREJEBCPPCJCRKIEACUZBKRVPKRBCIBQCARBJCVFCUPKRIOF KPACUZQEPBKRXPEIIEABDKPBCPFCDCCAFIEABDKPBCPFEQPKAZ
BKRHAIBKAPCCIBURCCDKDCCJCIDFUIXPAFFERBICZDFKABICBB
ENEFCUPJCVKABPCYDCCDPKBCOCPERKIVKSCPICBRKIJPKABI
Homework Equations
I know that...
Homework Statement
Below is an example of ciphertext obtained from an Affine Cipher. Determine the plaintext.
KQEREJEBCPPCJCRKIEACUZBKRVPKRBCIBQCARBJCVFCUPKRIOFKPACUZQEPBKRXPEIIEABDKPBCPFCDCCAFIEABDKPBCPFEQPKAZ
BKRHAIBKAPCCIBURCCDKDCCJCIDFUIXPAFFERBICZDFKABICBB...
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...
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)...
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 +...
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...
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...
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...
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.
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...
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...
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|...
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...
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...
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...
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...