Let S={xeZ:x+2=6(mod 15)}. Then S is

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

A:

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

A:

Q: I. Solve for the smallest value of x in the following linear congruences. 1. x = 3 mod 6 and x = 4…

A: To solve given linear congruences

Q: Let s={xEZ:x+2E6(mod 15)}. Then S is *

A: a countable set

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: 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: x is congruent to 43 mod 12. Then x is

A: x is congruent to 43 mod 12 implies x congruent 7 (mod 12). So, solution in the least residue system…

Q: 1. Let a, b e Z. Prove that if a = 0 (mod 5) and b = 2 (mod 5), then a² + b² = 4 (mod 5).

A:

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

A:

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: a. Let m be an integer such that m mod 5 = 2, what is 8m mod 5? b. Let n be an integer such that n…

A: Consider the given information: Let m be an integer such that m mod5 = 2. Then m=5k+2   where…

Q: Let n ∈ Z. Prove that if n^2 + 2n −4 doesnt equal 1 (mod 5), then 5 doesnt divide n.

A:

Q: For which p= 3 mod 4 is i a square in F»[i]? For which p = 3 mod 4 is i the square of an element in…

A:

Q: If n| m and a = b(mod m), then a = b(mod n).

A:

Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

A:

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: Given any odd prime p, prove that: 2(p−3)! ≡−1 mod p. (Hint: Start with 2(p−3)!≡x mod p. Then work…

A: Let p be any odd prime number.  So we start with  2p-3!≡x mod p Also we have p-2≡-2 mod pp-1≡-1 mod…

Q: The last two digits of 52044 is 44. If ab = 0 (mod 3), then a = 0 (mod 3) or b = 0 (mod 3). F 10. F…

A:

Q: Write the set of solutions of x=5 (mod 24) x= 17 (mod 18), if any as the solutions to a single…

A: We have a set of congruence as x ≡ 5 (mod 24)x ≡ 17 (mod 18) From the first equation we can write…

Q: suppose a,b ∈ ℤ. Prove that a ≡ b (mod 10) if and only if a ≡ b (mod 2) and a ≡ b (mod 5).

A: In Solution we use divisibility formula that  If a and b are integers, a divides b if there is an…

Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

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

A:

Q: Show that y = x (mod p) leads to -y = -x (mod p).

A:

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

A:

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

A:

Q: Show that: if a = b(mod n) and C = d\modn), then a +c =b+d\mod n)!

A:

Q: The order of 5 mod 17 is *

A: Since ϕ(17) = 16 Therefore order can be the divisor of 16 So the order can be 1,2,4,8 or 16

Q: 27 = x(mod 5) is true if x =

A:

Q: If mx = 1 (mod n) has a solution, then nx= 1 (mod m) has a solution. Select one: O True False

A:

Q: (b) Let x be an integer such that 0 < x < 8 and (10500 + 105000 ) = x (mod 9). Justifying your…

A: We have to find the required value of x .

Q: Let n E Z. Prove that 3n # 1 (mod 9).

A: Proof by contradiction: Direct approaches can be difficult (or impossible) to use to prove a…

Q: 10. Use Euler's Theorem to show that 5P(16), '=1 (mod 16). Show your work or explain your reasoning…

A:

Q: Write a formal proof for the following. For all integers a and b, if a ≡ 1 (mod 4) and b ≡ 3 (mod…

A:

Q: A sequence of pseudorandom numbers is generated as follows Xo = 4 x₁ = ( 6x₁-1 + 5 ) mod 13 if i > 0…

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: Let n E Z. Prove that if 5n2 +3 =0 (mod 8), then 4 n.

A: In this question  we will be using contradiction concept so assume it is true statement then we…

Q: Let a E Z. Prove that if a = 2 (mod 6), then a? = 4 (mod 12).

A: We will prove this by basic result of congruence

Q: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

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: If a and b are integers and n is a positive integer, prove that a mod n =b mod n if and only if n…

A: If a and b are integers and n is a positive integer,    Formula used: a≡b mod n if and only if  a-b…

Q: For D = {1, 2, 3, 4, ..., 99}, find a counterexample to the claim, "For all æ, y € D, if æ mod 3 = y…

A:

Q: The least positive integer x satisfying 22010 = 3x (mod 5) is

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 n be a positive integer, and assume that a and b are integers such that a ≡ b (mod n). Prove…

A: Looking at the definition of congurence we can easily do these two problems and we need some algebra…

Q: Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n =…

A: If a mod n =k and b mod n=j show that a+b mod n =j+k mod n. Definition: If x mod y=z, then y | x-z.…

Q: 3. Let d = (a, n) where n > 1. Prove that if there is a solution to ax = b (mod n), then d divides…

A:

Statements and Logic

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