An odd composite number is called a Carmichael number if a^(n-1) ≡ 1 (mod n) for all integers a with (a, n) = 1. Show that 1729 = 7 x 13 x 19 is a Carmichael number.
Show that the gap between two consecutive squarefree numbers can be arbitrary large.
Problem 5. Recall that a number n is called squarefree if it is not divisible by any square > 1. Show that the gap between two consecutive squarefree numbers can be arbitrary large. (Hint: Find a positive integer m such that m is divisible by 2^2, m + 1 is divisible by 3^2, m + 2 is divisible by 5^2, m + 3 is divisible by 7^2 an ...continues
Finding Two Prime Factors of a Number
The number n = 15744539 is a product of two prime numbers. Find these two prime numbers if it is also given that φ(n) = 15736560. You may only use elementary functions on your calculator (adding, subtracting, dividing, multiplying, taking the square root). Show your work.
Pointwise Operations and Characteristic Functions
Let U be a set, suppose f, g : U --> R are functions from U to the set of real numbers R, and Let a E R. Then f + g, fg. af: U --> R are defined by (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x) (af)(x) = a(f(x)) for all x E R. If a E R by abuse of notation we regard a as the constant function from U to R defined by a(x) = ...continues
Need help in proving the following. May need above assertion to prove. (See attached file for full problem description)
Pointwise Operations ( Composition and Inverse ) of Functions : Injective, Surjective and Bijective
3. Let f: A —> B and g: B —> C be functions. a) Suppose that f and g are injective. Show that g o f is injective. b) Suppose that f and g are surjective. Show that g o f is surjective. c) Suppose that f and g are bijective. Then g o f is bijective by parts a) and b). Show that (gof)^-1 = f^-1 o g^-1. Please do number 3. ...continues
Functions and Images : Injective and Surjective Functions
Let f: R --> R be the function defined by f(x) = x2 + 3x - 4 for all x E R. a) Determine wheter or not f is injective. b) Determine wheter or not f is surjective. c) Find f^-1({0,75/4}). d) Find f([0,1]).
Pointwise Operations and Characteristic Functions
Let U be a set, suppose f, g : U --> R are functions from U to the set of real numbers R, and Let a E R. Then f + g, fg. af: U --> R are defined by (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x) (af)(x) = a(f(x)) for all x E R. If a E R by abuse of notation we regard a as the constant function from U to R defined by a(x) = ...continues
Probability : Independent Events and Dice Rolls
1. Three dice, 1, 2, and 3, are rolled independently. • Event A12 is that dice 1 and 2 show the same number. • Event A13 is that dice 1 and 3 show the same number. • Event A23 is that dice 2 and 3 show the same number. (a) Are events A12 and A13 independent? (b) Are the three events independent?
Probability and Independent Events : Bayes Theorem
5. (Sudden death) The NHL has another season-long strike, but the owners and players reach an agreement in June which leaves time for a highly abbreviated season. They decide that fans want to see the Stanley Cup decided, and so they play only a sudden-death version of the seventh game of the final round of the playoffs. Her ...continues
