Q2a) For all integers a,b,c, prove or disprove each of the following: i) If a | c and b | c, then ab | c^2. ii) If a | (b+c), then a | b and a | c.b) Find in set builder notation the set of all positive integers a such that a≡137 (mod 5). c) Find the smallest positive integer a in the encryption function f(x)=(ax+7) mod 26,0≤x≤25,such that the function encodes the letter “H” by “C”.

Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

A:

Q: Find the identity element for the operation * on N defined by a *b=a+b+2, Va,beN al al a 240 does…

A: Let e be the identity element. Hence, it can be written that a*e=a .

Q: Find all x (mod 1999) such that x³ = 2 (mod 1999).

A: We use primitive roots to solve this problem.

Q: 1): For which values of n is n div 5 = -6? A): n = -30, -29, -28, -27, and -26. B): n = -30, -15,…

A: see below the answer and explanation

Q: Express each nonnegative integer a less than 15 as a pair (a mod 3, a mod 5).

A: The values of a could be, 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: Q// Let m = 5, 1- Write the Addition table (Cayley's Table). 2- Write the congruence classes of…

A: For m=5 , our set is Z5 .

Q: . Determine whether or not x³ + 2x + 1 and x4 + 3x - 2 are congruent mod (x² + 2x + 2) in Q[x].

A:

Q: Find the order of all element of Z8 under addition mod . Please brieflt explain and show all the…

A:

Q: Prove that the order of 3 modulo 19 is a divisor of 18.

A:

Q: 2 and an+1 Зап + 2. Find an explicit Suppose that a1, a2, a3, . .. is defined by a1 formula for a,…

A: Suppose that a1,a2,a3,...,...is defined by a1=2 and an+1=3an+2.find the explicit formula for an in…

Q: 1. For each solution: determine weather there is a a congruence a x² = -1 (mod 397) b.) x²+8x+14=0…

A: 1a) To determine whether the congruence equation x2 ≡ -1 (mod 397) has a solution, we need to check…

Q: 1. a) Determine whether binary operation is associative and whether it is commutative or not * is…

A: a) Given, a*b=a-b Now, a*b=a-bb*a=b-a So, a*b≠b*a So, * is not commutative Given, a*b=a-b Now,…

Q: Which of the following linear congruence has 2 incongruent solutions? (a) 3x = 9 (mod21) (b) 14x =…

A: to solve linear congurence , ax≡b(mod m) .........(1) the first stem is to reduce the congurence  by…

Q: SorvE THモ CONGROEモNしE 626 X Rエ mod5

A: As 5 is an odd number, by fermat's theorem, we have             x5-1 = 1 (mod 5)           => x4…

Q: Identify if the given statement is true or false: I. If a is an integer, the (a^2 is congruent to 0…

A: Given some statements. Check whether these statements are true or false.

Q: 2 2 Let A = Compute 412 - A and (412)A. 6-2

A:

Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: Show that the set QR/2) = {a + b{2 +c4:a,b,ce Q} with the usual actions of + and*and with the…

A: Given set is, Q23=a+b23+c43 Where a,b,c∈Q For sum of two elements,…

Q: - Let a, b, n be positive integers. Prove that if a = b (mod n), then (a, n) = (b, n).

A:

Q: et a, b, c be non-zero integers. Suppose that x, y are integers such that ax + by = c. Show that for…

A:

Q: Z+ as follows: For a, b € Z+, a*b = a - b + 1. Z+? Explain. (3.1) Define an operation * on Is * a…

A:

Q: Are the polynomials x³+2x +2 and x4 + 3x -2 equal to mod(x² + 2x + 2) in Q[x]?

A:

Q: For any integer a, prove that a = 0 or 1 (mod 3).

A:

Q: B- What is the multiplicative inverse of x¹5 in Z₂[x] (mod x² + x + 1)?

A:

Q: For any integer a, a37 = ? mod 1729

A: Option b is correct answer

Q: ind all solutions of the linear congruence: 9x = 5 mod (14)

A:

Q: Show that if b0 = 0, then (g^xm)≡ 1 (mod p). Q2: Show that if b0 = 1, then (g^xm) ≡ p − 1 (mod p).

A: Given: x is a number in the set {0, 1, . . . , 2t} x written as: x = b0 · 20 + b1 · 21 + b2 · 22 + ·…

Q: Q-2: a) Four married couples have bought 8 seats in the same row for a concert. In how many…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Statements and Logic

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