(a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each integer n, n³ = n (mod 3).

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: The statement a ≡ b (mod n) is called a congruence.

A: Given that  The statement a ≡ b (mod n) is called a congruence.

Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: Given 6x + 2 ≡ 101 (mod 7), find an integer z such that 2x ≡ z (mod 7)

A:

Q: (10) Find all solutions of the congruence 2 = 22 (mod 53). C

A: We shall be using primitive roots to solve this congruence.

Q: (7) (a) Find the order of a modulo 7 for a = 1, 2, 3, 4, 5, 6.

A:

Q: 1. Find all solutions (a) 2x + 3 = 7 (b) 2x + 3 = 7 to the following congruences.. (mod 9) (mod 20)

A:

Q: 1) Compute the order of a modulo 7 for a = 1,2,....6. Find all primitive roots modulo 7.

A: Order of the element

Q: Find all Pnterers x such that 5x=12 (mat) LoxE 4 (mad) 7x=16 (modlo) and 1o00s x3000

A:

Q: a) Find integers a and b such that a + b =a-b (mod 5).

A:

Q: (a) Prove that 2 and 3. are both primitive roots modulo 5.

A:

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 least residue of  -5 -46 modulo 14.

A:

Q: (2) Compute the order of a modulo 11 for a = 1,2,..., 10. Find all primitive roots modulo 11.

A:

Q: (b) Find 95! (mod 96).

A: We have to find the value of 95!(mod 96). Since 95!=95*94*93*…*8*7*6*5*4*3*2*1.

Q: 8) Prove that 2 is a primitive root modulo 53.

A: We have to Prove that 2 is a primitive root modulo 53.

Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

A:

Q: Show that there are no integers a, b such that a² + 2ab+26² = 100000000003. Hint. Use mod 4…

A:

Q: Which of the following satisfies the congruence 3x O O 2 T - 2 (mod 5)?

A:

Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

A:

Q: Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)

A: I have proved it by using defination of congruent.

Q: Use Wilson's Theorem to prove that for primes p congruent to 1 modulo 4 the quadratic congruence 1'…

A: Given The quadratic congruence x2≡-1(mod p) To prove  x2≡-1(mod p) has a solution if p≡1(mod 4)…

Q: 1. For each solution. congruence determine whether there is a J X² = -1 (mod 397) (397 is prime) CC…

A: We shall be using the result regarding the value of legendre symbol

Q: Solve the congruence 2² +1 =0 mod 133.

A:

Q: 3. Find all integer solutions to the pair of congruences (if any) x = 1 (mod 2) x = 2 (mod3) x = 3…

A: To Find: Integer solutions to the pair of congruences(if any) x≡1(mod 2), x≡2(mod 3), x≡3(mod 5)…

Q: Let A = {1, 2, ..., 12}. Let aRb mean a = b (mod 5). (a) Give [2]R. [3]R, and [5]R. (b) Give [2]…

A: A set of ordered pairs is referred to as a relation in the area of mathematics and computer science.…

Q: For n=195, by Fermat's theorem a-2 a) a' (modn) b) 1(modn) c) 0(modn) d) a(mod n)

A: Fermat's theorem

Q: Using Fermat's Little Theorem, solve the congruence 2. x425 + 4 · x108 – 3 . x² + x – 4 = 0 mod 107.…

A: Consider the provided question, 2·x425+4·x108-3·x2+x-4≡0  mod 107                  ......1 Using…

Q: 10.3 Prove that there exist infinitely many composite number n for which a-1 a(mod n)

A: Sol:- To prove that there exist infinitely many composite numbers n for which a(n-1) ≡ a (mod n), we…

Q: Let ø(n) be Euler's phi function. (a) Show that 1+2' + 22 + 23 + +26(15)–1 =0 (mod 15) but 1+3' + 32…

A: a) Let φ(n) be Euler's phi function we have to show   1+21+22+23+...+2φ(15)-1≡0mod 15 and…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.