Question 1
Let p be a prime, x be a positive integer which is not divisible by p, and y be the smallest positive integer where xy 1 (mod p). Prove that y j (p 1).
Question 2
Show that 169 - (2n2 + 10n 7), 8n 2 Z+.
Question 3
Let a and b be integers and m and n be positive integers. Given a b (mod m) and a b (mod n) where gcd(m; n) = 1 prove that a b (mod m n).
Question 4
Use mathematical induction to prove that for all positive integers k and n,
n
n(n+1)(n+2) (n+k)
jP
(k+1)
j(j + 1)(j + 2) (j + k 1) =
=1
Question 5
Let H0 = 1, H1 = 3, H2 = 5, and de ne
Hn = 5Hn 1 + 5Hn 2 + 63Hn 3
for n 3. Show by strong induction that Hn 7n for all n 0.
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• Regulations
1. You have to write your answers to the provided sections of the template answer le given.
2. Do not write any extra stu like question de nitions to the answer le. Just give your solution to the question. Otherwise you will get 0 from that question.
3. Late Submission: Not allowed!
4. Cheating: We have zero tolerance policy for cheating. People involved in cheating will be punished according to the university regulations.
5. Newsgroup: You must follow the newsgroup (cow.ceng.metu.edu.tr/c/courses-undergrad/ceng223) for discussions and possible updates on a daily basis.
6. Evaluation: Your latex le will be converted to pdf and evaluated by course assistants. The
.tex le will be checked for plagiarism automatically using "black-box" technique and manually by assistants, so make sure to obey the speci cations.
• Submission
Submission will be done via odtuclass. Download the given template answer le "the3.tex". When you nish your exam upload the .tex le with the same name to odtuclass.
Note: You cannot submit any other les. Don’t forget to make sure your .tex le is successfully compiled in Inek machines using the command below.
$ pdflatex the3.tex
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