Find the smallest integer m ≥ 1 such that the following statement is true: For any set S consisting of m positive integers, there exist a, b ∈ S (with a not equals b) such that at least one of a + b and a − b is divisible by 30.

Q: The greatest common divisor (gcd) of non-zero integers x and y is the the smallest positive integer…

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Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

Q: Let a,b, and c be integers. If gcd(a,b)=1 and both a and b divide c, then prove that a*b divides c.

A: Let a, b, and c be integers. If gcda, b=1 and both a and b divide c, then prove that a×b divides c.

Q: etermine whether the set of x-2x²,2+5x-x²x+x²} is linerly independent in P

A: First we use definition of linearly independent Set and we check given vectors are linearly…

Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: Prove that the product of two integers, one of the form 3k₁ +2 and the other of the form 3k2 +2,…

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Q: Let A be the set of integers that are multiples of 3 between 1 and 15 inclusive and B be the set of…

A: Multiples of 3 between 1 and 15 (inclusive) are {3,6,9,12,15}. Set of even natural numbers up to 20…

Q: Is a R a maximal deal

A: No, it is not necessarily true that aR is a maximal ideal in a commutative ring R with unity. In…

Q: Suppose that each of A, B, and C is a set and each of BnC, A–(BnC), A-B, A-C, and (A – B) n (A – C)…

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Q: Let a, b be integers. Define A = {n | n ∈ Z and a divides n} and B = {m | m ∈ Z and b divides m}.…

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Q: Prove that for any two integers a and b with b>0, there exist unique integers q (the quotient) and r…

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Q: Prove that for any integer a, one of the integers a, a + 2,a + 4 is divisible by 3.

A: We have to prove that one of the integers a,a+2,a+4 is divisible by 3.

Q: Prove that there are integers x and y such that 10x-13y=1

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Q: Use the pigeonhole principle to prove that any 6-element subset of {1,2,..., 10} contains a pair of…

A: This question from discrete mathematics and topic is pigenhole principle.

Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Prove that the following 2 sets are equal: {4k + 5 : k ∈ Z} = {4k + 1 : k ∈ Z}.

A: Two sets are equal if and only if They have the same element  We are using here if x €A. <=>…

Q: Prove this statement For all integers a, b, and c, if a ∤ bc, then a ∤ b

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Q: Finish the following set builder notations: The set of even numbers that are also perfect squares…

A: The set builder notation of “The set of even numbers that are also perfect squares where x is an…

Q: Prove by contradiction that for all integers a, b, and c, if a|b and a | c, then a |(b + c).

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Q: et ZZ be the set of all integers. Let R = {(a, b): a + b is divisible by 3}. a. Show that R is not…

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Q: Prove that for every integer n 2 0, gcd(Fn + 1, Fn) = 1.

A: Given: Fn is a Fibonacci sequence, with F0=1=F1. To prove: For every integer n≥0, gcdFn+1, Fn=1.

Q: Prove that, if a, b, and c are real numbers such that a <b< c, then |b| < max{Ja|, Jel}.

A: Consider the given: a, b ,c are real numbers  such that a&lt;b&lt;c

Q: Prove that the following statement is true for all positive integers m and n: m and n are…

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Q: let a,b, and c be integers such that if (a,c)=1 and (b,c)=1, then prove that (ab,c)=1

A: Introduction: If a and b are two numbers then the greatest common divisor of both the numbers is…

Q: Let n and m be integers. Prove that if m + n >= 100, then either m >= 50, or n >= 50.

A: Prove the statement by proving the contrapositive. The contrapositive of the given statement is as…

Statements and Logic

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