Determine whether the statement is true or false. Prove the statement directly from the definitions if it is true and give a counterexample if it is false. a. For all integers a, b, and c, if a | bc, then a | b or a | c. b. For all integers a and b, if a | b, then a² | b²

Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

A: we have 15a+20b =49 and a and b both are in R  we prove that a and b are not both integers.

Q: Let a and d be integers. Prove that if d|(8a+7)and d|(5a+4), then d|3

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Q: Prove the statement if it is true, and give a counterexample if it is not true. a. For every real…

A: Both the subparts are solved with explanation.

Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

Q: Assume that c and d are particular integers. Is 6c + 4d ² + 3 odd? 2

A:

Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: et a, b, c be positive real numbers. a² 6² c² a c

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Q: Let a, b, and c be natural numbers. Prove a ∙ (b + c) = a ∙b + a ∙ c.

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Q: a a + b Determine all values of a and b so that belongs to а 1 3 1 Span 1

A: We have

Q: let a b c and d be integers, prove if a|b and c|d then ac|bd

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Q: (b) Given integers a, b and c such that cla and cb then c|(ar + bs) for all r, s E Z. True False &…

A: Given that c|a and c|b for some given integers a,b and c The statement is c|ar+bs for all r,s∈ℤ…

Q: let a and b be integers. Show: if a * b is odd, then a is odd and b is odd

A: A number is even, if it is divisible by 2. A number is odd, if it is not divisible by 2. The odd…

Q: ) Recall that r > 0 for every positive real number r. Prove that if a and b are positive real…

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Q: Consider the following predicates defined on the domain of all real numbers. P(x): "x is positive"…

A: In mathematical reasoning, DeMorgan's law is very important. It is used in the case of compound…

Q: Identify the expression of a for which the following equation holds for every positive integers n…

A: