Module 2 – Congruence Arithmetic – Math 3303
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This is an 80 point assignment.
Homework rules:
Front side only. Keep the questions and your answers in order.
If you send it pdf, send it in a single scanned file. ()
If you turn it in personally, have the receptionist date stamp it and put it in my mailbox.
(651 PGH – 8am to 5pm)
Question 1
Write out the reasons for each step and piece of work to explain the proof that 341 divides on page 52 in the book. You may rearrange the steps so that the flow is better and you definitely need to put in explanations that are clear and complete. Write it up so I know you know what the authors are doing.
2. Make two mod 8 tables: one for adding and one for multiplying.
Solve for x
A.
B.
C. 2x 4
D. 5x 1
What’s the difference between problem C and problem D?
No more than two sentences of explanation, please.
3. Look up Pascal’s Triangle on the internet. Find 10 interesting facts – including those from our book – and record them in your own words along with the references.
4. Does 41 divide ? Yes, it does. Explicate the following steps.
Why did I make each choice?
Note that
Thus
So that:
Question 5
On page 46 in the text, the authors are discussing dividing a congruence by the same number on each side.
They rightly note that the congruence is preserved if the modulus, m, and a, the divisor, are relatively prime. Then they note that when m and a are not relatively prime the congruence relation is not always preserved and they note:
“It is not hard to develop a further rule for case (b) and we leave this to you”
Please do develop a further rule. Cite your sources if you find the rule elsewhere. Be sure your explanation, though, is in your words and that you understand it totally.
Note that sometimes dividing works and sometimes it doesn’t when a and m are not relatively prime. Start your work by showing 2 examples when it doesn’t work and 2 examples when it’s fine. Then find the rule that distinguishes the cases.
Question 6
Is 50, 625 – 1 divisible by 7?
Yes, it is. Prove this with a modular arithmetic style argument.
Question 7
In class, we looked at arranging the primes mod 6 and found an interesting pattern.
See about arranging the primes mod 12 and if you can spot a pattern there. Why does this work – or – why doesn’t this work?
Work neatly in Excel or on graph paper.
Question 8
On page 51 in the book, the authors state:
“Now, is the sum of all the numbers in the n’th row of Pascal’s Triangle.”
Prove this.
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