a) ⌊6.2⌋ ⌈3.4⌉ = ? b) 2 ⌊?⌋ = ? c) ⌊?⌋ = ⌈?⌉ for all a ∈ ℝ. True or False? Give a counterexample if false. d) ⌊?⌋ = ⌈?⌉ – 1 for all a ∈ R – ℤ. True or False? Give a counterexample if false. e) Is 80 congruent to 5 modulo 17? Why, or why not?

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A: We have to prove the given statement.

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A:

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A:

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A:

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A: First option is correct. gcd(a, m)=1.

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A:

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A:

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A: .

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A: Solution is given as below:

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A:

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A:

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A:

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A:

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A:

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A:

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A:

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A:

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A:

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A: Please see the below picture for detailed solution.

Q: 3.1 Prove: For all a and B in R, Ja + B] > | |a| – |B||. Hint: la + B] = |a – (-B)|-

A: Answer is in below steps.

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Q: (a) For integers a, b, q, r, prove that if a = bq + r then (a, b) = (b, r).

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