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: 2. Euclidean Algorithm Use the Euclidean algorithm to find gcd(12345, 678).

A:

Q: if a equiv b (mod n) , then a⁴ equiv b⁴ (mod n) for any positive integer 4.

A:

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

A: We use primitive roots to solve this problem.

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].

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

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Q: If x^677 ≡ 3 mod 2020, find x.

A: We have   x677≡3mod2020⇒x677-3=2020m   where m is integer different values of m we get different…

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

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

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Q: Test2

A: We know that2198=2128×264×24×22Also we have22≡4(mod 397)24≡16(mod 397)28≡256(mod…

Q: 12 = -3 (mod 5) Select one: True False

A:

Q: Draw the hyperbola with the equation: 3x2 − 2y2 + 6x − 8y = 6.

A: Given the equation of the hyperbola : 3x2-2y2+6x-8y=6.

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: 2. Euclidean Algorithm Use the Euclidean algorithm to find gcd(12345, 678).

A: We need to find the GCD(12345,678)=?

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: 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…

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Q: 4. Is the binary operation # defined on the set of real numbers commutative if a#b = a+ 2b + 4, Va,…

A:

Q: {m belongs to all integers| m^2=3}    Now list the elements in the set?

A: The given set is {m belongs to all integers| m2=3}.To list down the elements in the given…

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

A:

Q: 2. Encrypt the message STOP POLLUTION by translat- ing the letters into numbers, applying the given…

A: let's encrypt the message "STOP POLLUTION" using the given encryption functions one by one.For the…

Q: 1.Solve the following polynomial equations.  a)  2X3+1=x2+2X

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

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Q: x² - 2x = 8(mod19) X= (mod19)

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

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Q: Show that 18! ≡ -1 (mod 437).

A: We have to show that: 18!≡-1mod437

Q: Prove that a1 + a2 = b₁ + b₂ mod n.

A: We have to prove that a1+a2=b1+b2 (mod n).Given:a1≡b1 (mod n)a2≡b2 (mod n)This means there exist…

Q: Let а, b and c be integers. Use the method of proof by contradiction to prove that if a|b and a f c,…

A: We have to solve given problem:

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

A:

Q: of the apples is equal to 3 of the oranges. If there are 12 mor 5 In a fruit stall, - - annles than…

A: Fraction:  A small part of something.  In other words, numerical value that is not a whole number A…

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

A:

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…

Q: Find all integers sx such that 2x = 5 (mod 7).

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.

Statements and Logic

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