Fermat's "Little" Theorem states that whenver n is prime and a is an integer, d"-1 = 1 mod n a) If a = 133 and n = 157, then efficiently compute 133156 = 1 mod 157 b) If a = 25 and n = 97, then efficiently compute 25 · 2595 = 1 mod 97 Use the Extended Euclidean Algorithm to compute 25- = -31 mod 97. Then 259 = 8 mod 97. c) If a = 152 and n = 463, then efficiently compute 152464 mod 463 II

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: 1. How many solutions( mod 419) does x = 186 mod 419 have? (Show your work!) 2. How many solutions (…

A: Given: x2≡186 mod 101  To find: Number of solutions that exists. Method Used:  For any, xk≡b mod p,…

Q: 2) i) Show that 5 is a primitive root modulo 17. Find the all integers, less than 17, whose orders…

A: To show that 5 is a primitive root modulo 17, we need to demonstrate that it has the property of…

Q: Find all solutions of the congmonce 12X = 3 (mod 45)

A: The given problem is to find the all solutions for the given congruence equation.Given congruence…

Q: Establish that if a is an odd integer, then 2" а = 1 (mod 2" +2) for any n 2 1.

A:

Q: Solve for d, 19d ≡ 1 mod (37422000) using the "Extended Euclid's Algorithm"

A: The solution is given by using Extended Euclid's Algorithm as follows

Q: Find the smallest non negative value of D such that ED is congruent to 1 (mod b) when b=616…

A: As given in the hint the idea is to use the extended Euclidean algorthim to solve the congruence.…

Q: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

A:

Q: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: 237 (mod 10)

A: We use the definition of a (mod b) to find the value of 237 (mod 10).

Q: Prove that no integer congruent to 3 modulo 4 can be written as the sum of two squares.

A:

Q: Find the least residue of 99! (Mod 101)

A: Introduction: In number theory, Wilson's theorem is a very significant theorem. It states that for…

Q: Show that if x mod 8 = 1, then (3x −5) mod 6 = 4.

A: Solution

Q: 4) Find 25725 (mod 31)

A: