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: 74. Prove that for any real numbers a and b, |b − a|=|a − b|.

A: Given; a and b are any real numbers Consider, L.H.S=b-aWe can write it as,=b+-a=-1-b+a=-1a-b…

Q: a a +1 2² (² + ¹ + x² = 1 [(¹-x)² + (1 + x)² ] F1 2 2 2

A:

Q: Determine whether it is possible to express 1 as a liner combination of 1 = m· 44 + n· 91 ,. for…

A:

Q: Show that the square of any positive Integer is of the form 3m or 3m+1 for some integer m

A:

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

A:

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…

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A:

Q: . Homework. Let a and b be integers. If 7 | (17a +256), prove that 7 | (10a+11b).

A:

Q: Z/mZz Zm

A: We need to prove that Z / mZ is isomorphic to Zm .

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

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

A:

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

A:

Q: Fix an integer . Prove that the equation xn+xn-1+...+x-1=0 has a unique positive solution.

A: We have to prove that the equation xn+xn-1+...+x-1=0 has a unique positive solution. Explanation:…

Q: Let A be a set of n real numbers. Show that A+ A- A € (n).

A: As per the question we are given a set of n real numbers A and we have to show that : |A + A•A| =…

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

A:

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: let A. 2. . find A"

A: We have, A=1201To find An we will calculate A1,A2,A3,... and observe the pattern.

Q: d = (a, b, c).

A:

Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

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: Let A, B, and C are three sets, prove that Bx(AUC)=(B× A)U(B×C)

A: "Since you have posted multiple questions, we will answer the first question. If you want the…

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: Let a be a real number. For 1 A = 2 - fınd A² . a -2 – 2a 4 + 4a 2a? – 4 -2 | - None of the others…

A: First option is correct. See the attachment.

Q: is called the multiplicative identity. • 0 -1 • 9 • 1

A: The multiplicative identity is 1because multiplying a number by 1 leaves it unchanged . If a is any…

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A:

Q: of T2, T-:

A: Consider Tn={n, n2}, where n is an integer. The objective is to find the number of elements in each…

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Q: 2. Determine all values of x - y when x and y are integers such that x = 360 + y².

A: The given equation is. x2=360+y2x2-y2=360x+yx-y=360 Since it is given that x and y are integers, so…