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Number Theory

Number Theory

Find all the integers such that when the final digit is deleted the new integer divides the original one. Can you generalize this to deleting other digits?

Cryptology

Decipher the following CFQGE KAZEMF ZMAGVMC NMO VYSV which was obtained by a formula of the type y=kx (mod 26)

Problem solving.

A. Find 12/25 divide by 1/5 in two different ways. Explain your methods. b. Explain, using a diagram, how the following problems illustrate the two interpretations of division: partitive division ans measurement division. i. A road crew repaves 1 1/2 miles of road each day. How long will it take the crew to pave a 3/4 mi

Prime factorization

Rewrite in the simplest form. State the GCF(greatest common factor) of numerator and denominator in each case. 1. 34/85 2. 123123/567567 Find 5/9+7/12 using three different denominators. Give your answers as mixed numbers in lowest terms. State the LCM (least common multiple)of the denominators.

Prime numbers

Prove that if p is a prime number, then p divides , for all n≥p. Here [r] denotes the greatest integer ≤ r , for any real number r. Does this result generalize to a result about instead of p ?

Modular arithmetics

Show that z and iz have the same modulus. How are the graphs of these two numbers related?

Approximation of Integrals and Taylor Polynomials

Given ( INTEGRAL ln square(x)dx, as x from n to n+1 ) = ( INTEGRAL ln square (n+x)dx, as x from 0 to 1 ) = ( INTEGRAL [[ln(n+x) - ln(x) + ln(n)]square] dx, as x from 0 to 1 ), (a) Verify that ( LIMIT (n/ln(n)) [INTEGRAL (ln square (x)dx) - (ln square (n))] as n approach to the infinity ) = 1 (b) Compute LIMIT ((n square)/

Polynomials

Here is what the problem asks for: Give an example of a polynomial function f of degree 5 such that the only real roots of f(x) are -2,1,6 and f(2)=32. Show that your example works and leave f(x) in factored form.

Differentiation of Polynomials : Proofs

Please see the attached file for the fully formatted problems. Let g be a function which can be differentiated four times on the interval [-1,1]. Denote . 1) Show that when g is a polynomial of degree less than or equal to 3. 2) Let P be the interpolation polynomial of f at the points -1, , , 1. a) Show that . b

Primes and divisibility word problem

When the accountants for lose-a-digit Computer, Inc. had finished preparing their annual budget, they presented the final figures to the president, I.M. Smart. "It looks like a good year," he exclaimed. "The amount of the budget just happens to be the smallest number of cents (other than one cent) that is a perfect square, a per

Modern algebra

(a) Let G = GL(2,R) be the general linear group. Let H=GL(2,Q) and K= SL(2,R) = {A is an element of G: det (A) =1} Show that H,K are subgroups of G (b) Let p be a prime number and a is an element of Z. Prove that a^p ,is equivalent to, a mod p

Proof - Natural Numbers

Prove the following conjectures, that for all N in the set of Natural Numbers... Question continued in attachment.

Working with natural numbers and divisibility.

The natural number 28A9B consists of different numbers and A is not equal to 0. When the number is divided by 9 the remainder is 7 and when it is divided by 5, the remainder is 1. What is A-B=?