Use Fermat's theorem, show that if ged(a, 155) 1 then a1 = a (mod 15

Q: What is the number of solutions to the congruence in Z125? x³ + x² + 3 = 0 (mod 125)

A: As per the question we have to solve the following polynomial congruence equation : x3 + x2 + 3 ≡ 0…

Q: Number theory

A: Here, x6+1 is reducible in the field mod 2 (when x = 1). So,  the reduction of  x6+1 will be either…

Q: Suppose that p, q are distinct prime numbers, prove that p9-1 + gp-1 = 1 mod pq

A: We use Fermat's Little Theorem, to prove our claim.

Q: {X = I (mod 3) 2X= 2 (mod 5)

A:

Q: find x, where x= 9753 (mod 120) and -120<x<0

A: Ans: -87

Q: Suppose a E Z. Prove that if a³ 1 (mod 5), then a # 1 (mod 5).

A: In this question, we use the method of contradiction to prove the desired result.

Q: c) Using Fermat's Little Theorem, find 6142 mod 11.

A:

Q: Use Fermat's theorem, show that if ged(a, 155) = 1 then a121 = a (mod 155).

A:

Q: Find all integer solutions satisfying all the equations given. 5x =1 mod 39 3x =1 mod 25 4x =1 mod…

A: Concept :.   *   ax =6 mod n                           g (d (a,n) = d (let)…

Q: Find all values of x (using congruent notation) for 35x ≅ 3 (mod 89)

A:

Q: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: Find all solutions in {0, 1,2, ..., 12} to x26 + 15x13 = 12 mod 13.

A:

Q: Find the primitive roots (mod 17).

A:

Q: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

A:

Q: If m = 2* is a power of 2, explain how you could use repeated squaring to compute am (mod n) for all…

A: a) We have m = 2k. First we illustrate the method of repeated squaring to compute am (mod n). Then…

Q: a) How can an inverse of a modulo m be used to solve the congruence ax ≡ b (mod m) when gcd(a,m) =…

A:

Q: Find the remainder using Fermat’s Little theorem 452 mod 11.

A:

Q: 13. (a) Prove that for each integer a, if a 0 (mod 7), then a #0 (mod 7).

A:

Q: 5. Use the method of successive squaring to compute 3875 (mod 103) 1

A:

Q: X=4(mod nod 13) X2-7(mod 15)

A:

Q: Use Fermat's theorem, show that if gcd(a, 145) = 1 then a141 = a (mod 145).

A:

Q: Let a, n be positive integers. Knowing that 15a - 13n = 1, we deduce that the multiplicative inverse…

A:

Q: 23219 Compute 11 mod 3. Show your work and justify each step.

A:

Q: Use Fermat's Little Theorem to find (Show work.) a) 945 mod.23. b) 231002 mod. 41 c) (-9)933 mod. 13

A: Fermat's Little theorem states that " if p is a prime number and a is not divisible by p then…

Q: Use Hensel's Lemma to find a solution to the following congruence: a +3 x + 3 = 0 mod 54.

A: Let f(x) be a polynomial , p is a prime number . If x=a is a solution of congruence equation…

Q: Find an integer A so that: A79 = 7 (mod 277) Please solve step by step thanks!

A: Given an equation. Let's solve for A.

Q: Consider the congruence 9x   21 (mod 33). (a) Does 9 -1(mod 33) exist? Explain why or why not.

A:

Q: Prove by any method(contradiction/contrapositive or cases)   Prove that (a mod 2)(b mod 2)= (ab)…

A:

Q: Use Fermat's theorem, show that if gcd(a, 155) = 1 then a121 = a (mod 155). %3D

A:

Q: Find a number a such that 70a = 1 mod 17.

A:

Q: Find all x EZ221 such that x2 =97 (mod 221).

A: To find : all x∈ℤ221 such that x2≡97mod 221 Pre-requisite : Properties of Legendre symbol - 1. ap=1,…

Q: Determine all positive number replacements​ (less than the​ modulus) for the question mark that make…

A: Let the required number be x , where x&lt;1 .

Q: Use Hensel Lifting (twice) to find a solution to x² = 7 (mod 374).

A: According to the given information, it is required to use Hensel lifting method to find the…

Q: Use Fermat's theorem, show that if gcd(a, 145) = 1 then a141 = a (mod 145). !! imun m nunY

A:

Q: he smallest positive integer x such that x = 22 (mod 25), x =

A:

Q: Determine whether e(x)=15x+4(mod 26) and e(x)= 4x+5(mod 26) are invertibles or not. If yes, find the…

A: We are asked to determine if the given equation are invertible and if they are we have to find the…

Q: Using Fermat's theorem, find 3201 mod 11.

A: Euler's Theorem states that for any natural  number n, we  have  aϕ(n)≡1(mod n) where ϕ(n) denotes…

Q: Find a such that 5= 7* modulo 13.

A: Given: 5≡7xmodulo13. The objective is to find the value of x.

Q: Give an example to show that a ^2 ≡ b^2 (mod m) need not imply that a≡ b (mod m)

A: Here I listed given example of above statement.

Q: Q3/ Use Chinese theorem to solve the following congruence: X=2 mod 3 X=1 mod 5 X 2 mod 4

A: Since you have posted a multiple question ,I will solve the first question(i.e.Q3)  for you. To get…

Q: Q7 Compute 59^11 mod 13 Note: You should be able to do this using the modulo squaring method and…

A: 59^11 mod13

Q: (c) Use Fermat little theorem to show that (a + b)® = a® + ® mod prime p and any positive integer a.…

A:

Q: Show that if a = b mod m and b = c mod m, then we have a = c mod m. Use the definition of congruence…

A:

Q: Solve the linear congruence 5x = 3 (mod. 11). (Show all the work.)

A: The expression, a≡b (mod c)  implies that c divides a-b, that is when c divides a-b the remainder is…

Q: Use the repeated squaring method to calculate the following: 16^10 (mod 230)

A:

Q: Number theory

A:

Q: What is the modulus and argument of 3i.

A: Let z be a complex number then, z is written as z=x+ιywhere x and y are real numbers.We have to find…