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SCHEDULE EXERCISES ASSIGNMENTS BOOK ZOOM FIRE TIME-EDIT

Assignment 4 - Arithmetic

Please submit using the Fire system

Note: You are allowed to use our Haskell code to check your answers to Question 1 and Question 2. But you have to write down a fully detailed answer / proof in your submission.

Question 1

Consider the following two diophantine equations. For each one, explain whether or not it has a solution. If there are any solutions, give three different solutions.

(a) \(19x + 22y = 6\)

(b) \(22x - 55y = 14\)

Question 2

Consider the following three congruences:

\[ x \equiv 1 \; (\text{mod}\; 3) \\ x \equiv 3 \; (\text{mod}\; 5) \\ x \equiv 2 \; (\text{mod}\; 7) \]

(a) Find an integer x that satisfies all three congruences simultaneously. Use the Pulverizer, and the Chinese Remainder Theorem, and explain how you reached your solution, step by step.

(b) What solutions are there such that 0 ≤ x ≤ 200?

Question 3

For what natural numbers n can the following expression:

\[ 1 + 2 + 3 + \dots + n \]

be a prime number?

Example: For n=1, it is 1, so not a prime number. For n=2, it is 3, so a prime number. For n=3, it is 6, so not a prime number.

What can you say in general about all n? Make a concrete claim, and prove it.

Hint: Use the expression for arithmetic sums, and investigate how many divisors the result has.

Submission

Submission of this assignment should be done electronically through the Fire system.

The submission deadline is Wed, Oct 7, at 13:00. At this time, you should have submitted a serious attempt to solve the assignment. A serious attempt is either an answer you believe to be correct, or a partial answer plus a detailed explanation of what you have tried to come up with a full answer. An empty document is not a serious attempt.

After submitting, you have until Mon, Oct 19, at midnight to submit a completely correct version.

You can submit your answers (in Swedish or English) in any of the following formats:

  • A simple text file, ending in .txt. You can make text files in any text editor, and then upload it to the Fire system.

    (When using a text file, you may sometimes need to "invent" notation. Please be clear about what your notation means. A list of suggested notation for text files is provided by us. You may also use Unicode.)

  • A picture you have made in a painting program, ending in .gif, .jpg, or .png. You can use for example MS Paint, or Google Docs to make a picture, and upload it to the Fire system.

  • A PDF-file, ending in .pdf. You can make PDF files by for example using OpenOffice, LibreOffice, Microsoft Word, or Google Docs, and choosing export as PDF. Then, upload the PDF-file to the Fire system.

  • A scanned in document, ending in .pdf or .jpg. You can write your answer on a piece of paper, and use a scanner to scan it in and convert to a PDF-file. Or you can use a camera or your mobile phone to take pictures of the papers, and upload these to the Fire system.

If you submit multiple files, please name and/or number them such that the order in which we should read them is obvious. You can also write a text file, and have it refer to pictures that you upload separately.