Final Exam

COURSE: MATH 2170

DATE & TIME: April 28, 2021,

Math数学网课代考 Given that 2 is a primitive root modulo 131, find the solutions. Note that 131 is prime and 2 is clearly a solution.

[5]1.Leta, g > 0 be given integers. Prove that integers x and y exist satisfying x + y = a and (x, y) = g if and only if g | a. (Quest Vers: 72) Math数学网课代考

[9](a) Use the Euclidean Algorithm to find (1792,1486). Show all steps in the algorithm.

(b)Computethe inverse of 743 modulo 896. Answer should be in the least residue.

(c)Findall integer solutions to 1792x + 1486y = (1792, 1486). Answer should be in the least residue.

(Quest Vers: 72)  Math数学网课代考

[5]3.Solve the system using the Chinese Remainder Theorem Method taught in class.

x ≡ 5 (mod 79)

x ≡ 10 (mod 14)

(Quest Vers: 72)

[5] 4. What are the last two digits of 13³²⁵? Show all steps using techniques from this class. Justify any theorems used. (Quest Vers: 72)  Math数学网课代考

[8] 5. Let f (x) = x⁵  2x⁴  2x³ +9x²  9x+3. It’s given that 1, 2, 4 are the solutions to f (x)  0 (mod 5). Using the methods taught with Hensel’s Lemma, determine the solutions to f (x) ≡ 0 (mod 25). (Quest Vers: 72)

[4] 6. Determine the order of 2 modulo 113. Note that 113 is prime. Show all calculations done. (Quest Vers: 72)  Math数学网课代考

[7] 7.  (a) How many solutions are  there to x⁵ 32 (mod 131)? Note that 131 is prime and 2 is clearly a solution. Explain your answer.

(b) Given that 2 is a primitive root modulo 131, find the solutions. Note that 131 is prime and 2 is clearly a solution. You may leave your answers in exponent form.

(Quest Vers: 72)

[7](a)Note that 33614 = 2   7⁵.  Is there any m such that Z×33614  ∼= Z_m? If so, find m with complete justification. If not, explain why not. Math数学网课代考

(b)  Is Z×35  ∼= Z₂₄?  Explain why or why not. (Quest Vers: 72)

[6] 9.  Given that 533 = 13   41, how many solutions are there to x² 139 (mod 533)? Justify your answer. (Quest Vers: 72)

[5] 10.

Find f (p) such that d|n µ(d)φ(d) = Π_p|nf (p). Completely justify your answer.  (Quest Vers: 72)

[6] 11.Find, with proof, all Pythagorean triples (x, y, z) with x odd, y even, x, y, z > 0, andx, y, z relatively prime which contain 65. (Quest Vers: 72) Math数学网课代考

[10] 12.Consider the equation x² = y³ + 23. (Quest Vers: 72)

Supposethere is a solution (x, y) to the equation, show that 3 ‡ y

Supposethere is a solution (x, y) to the equation, show that x is even and y 1 (mod 4).

Show there is no solution to the equation. (Hint: What happens if you add 4 to bothsides?)