QUESTION 11. Reformulate Equation (2.1), removing the restriction thatais a nonnegative integer. That is, letabe any integer.QUESTION 21. For each of the following equations,find an integer x that satisfies the equation.a) 5x≡4 (mod 3)  answer?b) 7x≡6 (mod 5) answer?c) 9x≡8 (mod 7) answer?QUESTION 31. Determine the GCD of the following;a) gcd(24140, 16762) answer?b) gcd(4655, 12075) answer?QUESTION 41. Using Fermat’s theorem to find a numberxbetween 0 and 28 withx85congruent to 6 modulo 29.(You should not need to use any brute-force searching.)QUESTION 51. Use Euler’s theorem to find a numberabetween 0and 9such thatais congruent to 71000modulo 10.(Note:This is the same as the last digit of the decimal expansion of 71000).QUESTION 61. Prove the following:Ifpis prime, then φ(p1) =pi-pi-1. Hint: What numbers have a factor in commonQUESTION 71. Six professors begin courses onMonday,Tuesday,Wednesday,Thursday,Friday,and Saturday,respectively, and announce their intentions of lecturing at intervals of 2,3,4,1, 6,and5days,respectively.Theregulations of the university forbid Sunday lectures (so that a Sunday lecture must be omitted).When first will all six professors find themselves compelled to omit a lecture? Hint: Use the CRT.2. Welcome to Week 2.3. This week we are going to be looking at Chapter 2 in our text and delving into number theory, whichis pervasive in cryptographic algorithms.4. The first three sections of Chapter 2 introduce basic concepts from number theory that are needed for understanding finite fields; these include divisibility, theEuclidianalgorithm, and modular arithmetic. Going over these sections now will set a foundation, but feel free to return and revisit these sections as we get ready to tackle Chapter 5 on finite fields.5. Sections 2.4 through 2.8 discuss aspects of number theory related to prime numbers and discrete logarithms. These topics are fundamental to the design of asymmetric (public-key) cryptographic algorithms. Again, you may want to return to these sections as we approach Chapters 9 and 10.6. In addition to your reading assignment for this week, we have a set of exercises to help reinforce the applications of the principles that we are discussing. These exercises come out of the problems at the end of the chapter, and you can feel free to work them out of your book if you are more comfortable. Once you have the solutions, just insert your answers into the Exercise 2 quiz and submit. These are set up as quizzes for simplicity and to allow you to continue to rework as needed up to the due date on Sunday. There is no time limit and you can rework your answers as many times as you like up to the due date.7. If you have any questions, please reach out to me and we will work through them.Lacture Chapter 2 this test form to submit your answers for this week’s exercises. Answers may be submitted and reworked multiple times leading up to the due date.

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