
(1) Define what an integral domain is. (2) Find all solutions to r? + 5x +...
Please show question 1 (all parts). Thank
you!
1. Using the Euclidean algorithm to find the ged of following pairs. Write down the ged as a linear combination of given pairs (a) 524 and 148 in Z (b)33 + 2r +1 and 2 +1 in Zs[] (c) 3 +2r +1 and 1 n Z[] 2. Compute 42001 in Z5 3. Use principal of induction show that 10" 1 mod 9 4. Show that every odd integer is congruent to 1...
Prove that x=+,- 1 are the only solutions to the equation x^2=1 in an integral domain. Find a ring in which the equation x^2=1 has more than two solutions.
Please solve the above 4 questions.
1. Using the extended Euclidean Algorithm, find all solutions of the linear congruence 217x 133 (mod 329), where 0 x < 329 (Eg. if 5n, n 0,. ,6) 24 + 5n, п %3D 0, 1, . .., 6, type 24 + x< 11 2. Find all solutions of the congruence 7x = 5 (mod 11) where 0 (Eg. if 4,7 10, 13, type 4,7,10,13, none. or if there are no solutions, type I 3....
12. Find all solutions with 0 <I<27: sec r = -2 13. Find all real solutions: sin r 2 14. Find all real solutions: 3 tan (3x) + 1 = 0
B3 a. Solve for x in this equation: 2x + 11 = 2 (mod 4). b. What are the sets of units and zero divisors in the ring of integers modulo 22? (Specify at least the smaller set using set-roster notation.) c. Find a formula for the quotient and the exact remainder when 534 is divided by 8. Hint: find the remainder first by modular arithmetic. Then subtract the remainder from the power and divide to find the quotient.
1. Find all solutions to this trigonometric equation. Use radians. sin(3z-.15) 9128 2. Find all solutions to this trigonometric equation. Use radians or degrees, your choice. tan (2r)-10 3tan(2r) 3. Solve the triangle whose three sides have lengths a 4, 8, c =11. a- 4 c 11 4. Solve the triangle where one angle α 30°, the opposite side 4, and one of the other sides is 7 (make it b). a α 300 b=7
please answer all the questions.
question 1 to question 5
Given an integral domain R we define the relatic n~on Rx (R (0]) by (a, b)~(c, d) means ad bc. We also define the following operations on R x (R\o) (a, b) + (c, d) (ad + be, bd) and (a, b) (c,d) (ac, bd). 1. Prove that ~ is an equivalence relation. 2. Prove that ~is compatible with +and . (Therefore, ~is a congru- 3. Conclude that the following...
Please help me with understandable solutions for question 6(a), 7,
8 and 10. ( Use Chinese remainder theorem where applicable).
78 CHAPTER 5. THE CHINESE REMAINDER THEOREM 6. (a) Let m mi,m2 Then r a (mod mi), ag (mod m2) can be solved if and only if (m, m2) | a1-a2. The solution, when it exists, is unique modulo m. (b) Using part (a) prove the Chinese remainder theorem by induction. 7. There is a number. It has no remainder...
Problem 1 Use the Chinese remainder theorem, find all integers x such that: (20 pts) x = 1 (mod 5) r = 2 (mod 7) x = 3 (mod 9) I= 4 mod 11) Answer,
Find all solutions using exact values cos 3x + cos 5x = 0 sec 2.c sec 6.r = 0.