QuestionQuestion

I. Find the following (Show your complete solution.)
1. The remainder when 5¹⁴⁰ is divided by 11. [15 points]
2. All positive integers n for which 11 ≡ 4 (mod n). [10 points]
3. The general solution to 15x ≡ 11 (mod 73). [10 points
4. All integers between 1000 and 2000 that leave a remainder of 2 when divided by
5, a remainder of 4 when divided by 7 and a remainder of 1 when divided by 11. [15 points]
II. Word Problems
5. An all-you-can-eat restaurant charges 750 pesos for adults and 650 pesos for senior citizens. Children eat for free. At the end of the day, the restaurant had collected 25,000 pesos. What is the largest number of people who could have eaten there that day? What is the smallest number of people? [15 points]
III. Proving
6. Let a, b € Z such that (a,b) = 1. Prove that (a + b, a² - ab + b²) = 1 or 3. [10 points]
7. Let Nn be the integer whose decimal expansion consists of n consecutive ones.
For example, N2 = 11 and N4 = 1, 111 . Show that (Nn,Nmm) = N(n,m)- [10 points]
8. A number N is divisible by 7, 11 and 13 if and only if the alternating sum and difference of blocks of three ddigits is divisible by 7, 11 and 13. [15 points]

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:
    Solution.pdf.

    $60.00
    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Number Theory Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats