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 not asymptotic.

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)}.