Let k and n be two positive integers, and assume that n is odd. Prove that there exists an integer a such that a32 = (n + 1)³ (n*).

Q: {p € Z :x |p}U{p €Z:y|p} C {p E Z : n | p}.

A: Number theory

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A: Expand 1+xn 1+xn=C0n(1)nx0+C1n(1)n-1x1+C2n(1)n-2x2+.......+Cnn(1)0xn=∑r=0nCrnxn Expand 1-xn…

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A: Given; Pn : 1/2 + 1/4 +1/8 +...+1/2n =(2n -1)/2n

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Q: Prove that there do not exist integers m and n such that 12m + 15n = 1.

A: Proof by contradiction. Assume the given statement is true.

Q: Find a positive integer m > 3 such that there are at least 3 integers n in Zm such that n2 = 1.

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Q: Prove that the following: 2"*(-1)"+1 (- 1)"(2)" = for all real non-negative integers (n) 3

A: Given :- The discrete problem -1n 2n = 2n+1 -1n + 13 for all real non-negative integers n.

Q: Let n be a positive integer. Prove that (:) () (:) 1. 1 + 2. 2 +n n2"-1

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Q: Let a, b € Z with a | b. Prove that an | bn for every positive integer n.

A: Given that           a, b∈ℤ with a|b. To prove an|bn  , we shall use mathematical induction.

Q: 14. Let (a, b) = 1 and r = y® for some integers r and y. Then prove that there exists an %3D integer…

A: As per policy, we are solving only the first Question please repost 2nd Question to get its…

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A: What is Divisibility: Divisibility is the base of number theory. For two given integer with one…

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A: (a) ans:- 1,5 (b) ans:- 2,3 &amp;6 For detail...go through the solution

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Q: 18. a. Prove that for every integer a, if a' is even then a is even. b. Prove that V2 is irrational.

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A: if gcd (a,b) =d then there exists integers m and n such that d = a m + b  n  and d is the smallest…

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Q: .3. a. Prove that for every integer a, if a' is even then a is even.

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Q: Prove by contradiction that 6n + 5 is odd for all integers n.

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

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