Let C={x €Z: x =7 mod 9} and let D={x € Z,x = 1 mod 3). a) List five elements of C and five elements of D. b) Is Cc D? Offer a proof or counterexample. Keep in mind that if a = b mod n that a - b =ng for some q EZ. c) Is DCC? Offer a proof or counterexample.

Q: Show that if a and b are integers and m and n are positive integers such that n|m, then If a = b(mod…

A:

Q: Let x E Z and y E Z. Prove the following statement: If y = 2 mod 4, then xy 1 mod 4.

A:

Q: Prove that if ca = cb (mod n), then a = b (mod n/d) , where d = gcd(c,n). %3D

A:

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

A:

Q: 2=1 or 5 (mod 6) , and if there exists a A-fold triple system of v, prove that v = 1,3 (mod6)|

A:

Q: Show that x2 = Y mod p has

A:

Q: Use the table in VII to give a formal proof of the following. For all integers a and b, if a = 1…

A:

Q: Let A = {r € Z | r mod 15 = 5} and B = {x € Z| r mod 3 = 2}. Give an outline of a proof that A C B,…

A: Suppose we have two non-empty sets A and B . If x∈A then x∈B for all x∈A then A⊆B . If x∈B then x∈A…

Q: consider the statement: for all n∈N , x∈Z , x^2 ≡ 1 mod n (        ) x ≡ 1 mod n or x≡ -1 mod n a)…

A:

Q: Prove that if ax = b (modn) then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x =…

A:

Q: The linear congruence ax = b(mod m) has a unique solutionif and only if: Od(a,b)=1 @d(a,m) = 1…

A: gcd(a,m)=1

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

A:

Q: The linear congruence ax = b(mod m) has a unique solutionif and only if: gcd (a,m) =1 gcd(a,m) =1 o…

A: First option is correct. gcd(a, m)=1.

Q: If a = b (mod m) and a = b (mod n), and gcd (m,n) = 1, then a = b (mod mn) Select one: O True O…

A:

Q: Let a, b, c and meZ, and m 2 2 1. Prove: if a =b (mod m), then GCF (a, m) = GCF (b, m) %3D m 2.…

A:

Q: The linear congruence ax = b(mod m) has a unique solutionif and onlyif: @cd(a,b) = 1 cd(a,m) =1…

A: Second option is correct.

Q: 2. In Section 3.1, we defined congruence modulon where n is a natural number. If a and b are…

A:

Q: The linear congruence ax = b(mod m) has a unique solution if and only if: god(a,m) =1 @cd(a,b)=1…

A: Solution: The linear congruence equation is ax≡bmod m

Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

A:

Q: Let a, b, c, m be integers with m > 2. Prove that if ac ≡ bc mod m and d = gcd(m, c) , then a ≡ b…

A:

Q: 4. Exercise $2.5 #58a. Let a and b be integers not divisible by the prime mumber p. (a) If aP = mod…

A: Given the data

Q: 4. Using your work from the earlier parts or independently find: (a) mod 8 (b) + mod 23 17 (c) mod 7

A: Given

Q: 9. Find all integers x, 0 < x < n, satisfying each of the fol- lowing congruences mod n. If no such…

A: Q (d),(g) asked and answered.

Q: The linear congruence ax = b(mod m) has a unique solutionif and onlyif: gcd(a,b)= 1 gcd(a,m) #1 O…

A: Gcd(a,m)=1

Q: Give an example to show that a? = b? (mod n) need not imply that a = b mod n). fa=b(mod n) prove…

A:

Q: Prove the following results. (I). If n = 2(mod 4), then n is the sum of three squares of integers.…

A:

Q: Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can…

A:

Q: Let S := {1< a<p - 1: () = 1} . Ifp = 1 mod 4, show that Σ p(p – 1) 4 rES

A:

Q: Prove directly from the definition of congruence modulo n that if a,c, and n are integers,n >1, and…

A: Let a, c and n be integers such that n&gt;1. We have to prove that if a≡cmod n, then a3≡c3mod n…

Q: (4) Let p be a prime number. Then r = 1 mod p has exactly two solutions modulo p.

A: 4) To prove, x2≡1mod p has exactly 2 roots such that p is a prime number.

Q: If a+c=b+c (mod m) then a =b (mod m) Select one: True False

A:

Q: Let c, d e Z and m E N. For each statement decide whether it is necessarily true, or whether it can…

A:

Q: Prove directly from the definition of congruence modulo n that if a, c, and n are integers, n > 1,…

A:

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A: Given below an step by step solution

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A: We have to find few terms of the given sequences.

Q: The linear congruence ax = b(mod m) has a unique solutionif and only if: gcd(a,b) 1 gcd(a,b) = 1…

A: gcd(a,m)=1

Q: 3) If the gcd(z,n) = d and xz = yz mod n then: Show that x = y mod

A:

Q: Prove that if a = b mod n and c= d mod n, then ac bd mod n.

A: Given if a≡bmodn and c≡dmodn We have to show that ac≡bdmodn. Now, a≡bmodn…

Q: Show that for x and y integers, and m a positive integer x = y mod m implies x = y mod m.

A: This is a problem of Number Theory, Modular Arithmatic.

Q: 4. Given the Ecryption code R = f(S) = (3S+5) (mod 26), find the decryption code ( i.e. %3D find…

A:

Q: 5. Let C = {x ∈ Z|x ≡ 7( mod 9)} and D = {x ∈ Z|x ≡ 1( mod 3)}. (a) If C ⊆ D? Justify your…

A:

Q: Given that a, b, c,m are integers with m > 0. If a = b (mod m) then prove that ас3 bе (mod m)

A: we have to use the only definition of congruence and the definition of divisibility

Q: Let S:= {1<a<p – 1:() = 1} . Ifp = 1 mod 4, show that p(p – 1) 4 rES

A:

Q: Prove the following statements using either direct or contrapositive proof. 1. Let x,y eZ. If x+y is…

A: Please see the below picture for detailed solution.

Q: Let P be a prime, if ac ≡ bc (mod p) and p does not divide c, then a ≡ b (mod p) Select one: True…

A:

Q: (b) Solve the set of congruences 2x =1 (mod 5) x=3 (mod 4)

A: Note:- We’ll answer the first question according our protocol. Please submit a new question…

Statements and Logic

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