(b) Find the 2 elements x in Φ(289) such that x2 = 77 (mod 289) (Hint. First solve the congruence modulo 17.)
Find all solutions to the congruence x2+ x+ 1≡0 mod 91. (Hint:factor the modulus, use trial and error to find the solutions modulo the factors, and the CRT to combine the results into solutions to the original equations.)
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.
15. Show that 716-1 (mod 17) and use that congruence to find the least non- negative residue of 7546 modulo 17
13. Solve the congruence: 341x = 2941 (mod 9). v Hint First reduce each number modulo 9, which can be done by adding up the digits of the numbers.
Arrange the steps in the correct order to solve the system of congruences x 2 (mod 3), x 1 mod 4). and x3 (mod 5) using the method of back substitution Rank the options below Thus, x= 31.2 - 3/4 + 1)2 - 120+5 We substitute this into the third congruence to obtain 12.5 13 mod 5), which implesu li imod 5) Hence, w5v4 and so x 12.5 - 12/5 + 4) - 5 - 60v. 53, where vis an...
Problem 2. Solve the congruence equation x( 12 mod 143 Problem 2. Solve the congruence equation x( 12 mod 143
Which of the following equations have solution? justify your answers a) x2 =3 (mod 137 ) this mean x to the 2nd power congruence to 3 mod 13 to the 7 power b) x3 =4 (mod 115.239) this mean x to 3rd power congruence to 4 mod 11 to the 5 th power times 23 to the 9th power c) x7=2(mod49) this mean x to the 7th power congruence to 2 mod 7 to the 2nd power d) x7 =...
(a) Solve the simultaneous congruences p = 1 (mod x – 3), p = 7 (mod x – 5). (b) Find the total number of monic irreducible polynomials of degree 5 in Fr[c]. (c) Find a primitive root modulo 52020. (Make sure to justify your answer.) (d) Determine the total number of primitive roots modulo 52020.
7. For all values of a and b modulo 2, solve 2 +b=0 (mod 2) ar 7. For all values of a and b modulo 2, solve 2 +b=0 (mod 2) ar
Problem 3. Use the Chinese Remainder Theorem to find all congruence classes that satisfy x2 = 1 mod 77.