1) If a and b are given integers, not both zero, show that the set T =(ax+by: x,y are integers} is the set of all multiples of d = gcd (a, b). 2) Find the gcd (54, 39)

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A: This is a problem of set theory and cardinality.

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Q: 6) find all the and non residues (mods) incongruent quadratic residues

A: It is given that Incongurant quadratic residues.

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A:

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Q: 4. Prove that C, = ,C4 (Hint: Algebraically show that the Left Side = Right Side)

A: We know that  Cr n= Crn = Crn = (n)!(r)! × (n-r)!

Q: Abstract Algebra

A: To find non-negative representations under the given conditions

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A: We will answer the first question only as we don't answer multiple questions at a time. Please…

Q: 15. Let A and b Show that the cquation Ax b does nol have a solution for all possible b, and…

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Q: 1. Show that the distance between two numbers in the real number line, given by d(x,y)=|x-yl,…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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A: The objective is to prove that every set ∑of sentences that has finite models of all odd sizes must…

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Q: Prove that ∀n ∈Z, gcd(7n + 1, 15n + 2) = 1 *use only the definition of divisibility and gcd please!

A: we want to prove that   gcd(7n+1 , 15n+2)=1

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A:

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A: The product of any two even integers is a multiple of 4.

Q: السؤال 2 2 The set P of positive integers has maximal element صواب ihi

A: Use th Archemedian property of set theory.

Q: Find all integer x satisfying the condition x = gcd(5x + 2 3r) lustify vOur ancu

A: The complete solution is in given below

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A: Solving

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A: We have the given set is S=x:x=a+ib, a,b∈Z where Z is the set of integers. First we prove that S is…

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A: Answer is mentioned below

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Q: If a and b are integers with gcd(a, b) =d Then gea (-) -- gcd

A: False

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A: To simply the sets in roaster form

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A:

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Q: (a) Determine whether the permutation 42135 of the set {1, 2, 3, 4, 5} is even or odd. (b) Find all…

A: The given permutation is 42135. The given permutation is a cyclic form of the set {1,2,3,4,5}. The…

Q: Prove that the set of even integers { . . . . -6, -4, -2, 0, 2, 4, 6 . . . . } is countable by…

A:

Q: Show that if seven integers are selected from the first 10 positive integers, there must be at least…

A: We can solve this question by pigeonhole principle

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A: Two integers are relatively prime when there are no common factors other than 1. This means that no…

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A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: Let (a,b)=1 Show that (a+b, a-b)=1 or 2. When is this GCD 1? When is 2?

A: Proof:It is given that gcd(a,b) = 1. So ax + by = 1.Let gcd(a + b, a – b) = d, then d|( a + b) and…

Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

A:

Q: Compute: gcd(51, 42) Find a pair of integers x and y such that 51x + 42y = gcd(51, 42) (x, y) = (…

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