Discrete mathematics proof

(See attached file for full problem description) --- Let d,m and n be positive integers with m>1 and m≡ 1 (mod d), let n= c0+mc1+m2c2+m3c3+…+mrcr be the base=m expansion of n, and let f = c0+c1+c2+c3+…+cr Prove that n is divisible by d if and only if f is divisible by d. ---

RSA encryptions

(See attached file for full problem description) --- Consider the RSA encryption system given by p=43,q=59, and e=13 i) Find d such that ed ≡ 1 (mod (p-1)(q-1)) ii) Decode the message : 1552 2069 1178 1637 1975 Using the convention A = 00, B = 01, …, Z = 25 ---

Discrete mathematics proofs

1) Prove that all integers a,b,p, with p>0 and q>0 that ((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q Or give a counterexample 2) prove for all integers a,b,p,q with p>0 and q>0 that ((a-b)mod p) mod q=0 if and only if (a mod p) mod q = (b mod p) mod q Or give a counterexample. 3) let p and ...continues

Extended Euclidian Algorithm Proofs

(See attached file for full problem description) --- Given positive integers a and b, the extended Euclidian algorithm constructs sequences qn, rn, sn and tn, which are defined recursively as follows: q0=0, q1=0, qn= q└ rn-2/ rn-1 ┘ for n>=2; r0=a, r1=b, rn= rn-2 - qnrn-1 for n>=2; s0=1, s1=0, sn= sn- ...continues

Hasse Diagrams

(a) List the ordered pairs that belong to the relation. My ANS: (a,a),(b,a)(b,b),(c,a),(c,b),(c,c),(d,a),(d,b)(d,d) (b) Find the (boolean) matrix of the relation. < my answer file attached as Mr.jpg> I did review another answer posted along the same question, but uses a different shape. I think my matrix is correct bas ...continues

Discrete mathematics Growth Functions big oh big theta

(See attached file for full problem description with proper symbols) --- 1.The functions f and g from the set of real numbers to the set of real numbers are asymptotic or f~g if lim x∞ f(x)/g(x)=1. Let f(x) = log (1+x2) and g(x) = log x. Prove that f(x) is g(x)) but fand g are ...continues

Euler's theorem applied to large integers

Consider Euler’s theorem: If m is a positive integer and a is an integer relatively prime to m, then a^phi(m)≡1(mod m) Use this theorem to show that if a is an integer relatively prime to 32760 then a^12≡1(mod 32760). Symbols better shown in file (attached).

Divisibility Tests and Rules and Quotient Polynomials

Let n be a positive integer a) prove that n is divisible by 5 if and only if it ends with 0,5 b) prove than n is divisible by 11 if and only if the alternating sum of its digits is divisible by 11 c) find a similar criterion for divisibility by 7 and prove it .

Normal distribution

Please see the attached file "Q11.5.doc" for the problem statement. suppose that w is normal and has expected value a and variance sigma^2. so the expected value of u = exp(w) is exp(a + 0.5sigma^2). What is the variance of u: var(u)?

Recursive definitions

(See attached file for full problem description) --- Give a recursive definition of a) the sequence {an}, n=1,2,3,…if i. an = 1+(-1)n ii. an = n2 b) of the set of ordered pairs of positive integers S = {(a,b) | a є Z+, b є Z+, and 3 |(a+b)}.