Let n ≥ 1 be an integer. Show that in any set of n consecutive integers, there is exactly one that is divisible by n. Hint: Express the smallest integer in the set as q ⋅n+r, where q, r are integers satisfying 0 ≤ r ≤ n−1.

Q: Abstract Algebra. Please explain everything in detail.

A: To prove (or disprove)  the possibilty of required representations under the given conditions

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A: Given: b measures a To proof: -b measures a

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A: Introduction: In Mathematics, the term "integer" was adapted from Latin. Integer means complete or…

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A: using :- if a|b then a=bq , where q is unique integer.

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A: our objective is to find A intersection B

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Q: 2. For all real numbers a, b and c, prove that if a <b <c then a+b+c a < 3 < C.

A: given that a, b, c are real numbers we have to prove that if      a&lt;b&lt;c    then…

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

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Q: x]+n x+n m m

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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

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Q: 4. Let R be the set of all real numbers, xER and n E N such that n 2 2. If (1+x) > 0, prove that P,…

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A: We prove it by contradiction.

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A: (xy)n=xnyn is true.

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A: Introduction: Divisibility is the base of number theory. For two integers a and b, if there exists…

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A: Let a = bq + r   … (1)Now,  let d1   = gcd ( a,b) then ⇒d1  | a  and   d1  | b ⇒d1 | a - bq   and d1…

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A: See the detailed solution below.

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