Take Home Exam 3 Solved

Description

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 x^y 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

H_n = 5H_{n 1} + 5H_{n 2} + 63H_{n 3}

for n 3. Show by strong induction that H_n 7ⁿ for all n 0.

• 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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