Let i be a positive integer and A; =] - i+1,1 [. Then N A= None of these O 10,1] O 10,1[ [0,1[

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

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Q: Prove thq+ for all integer a, b, and c if 9+b of a and k is even.

A: Introduction: Even and odd numbers are also referred to as: Even Number: An even number is one that…

Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: 23. Prove that for any integer a, 9/ (a² – 3). -

A: Prove that for any integer a, 9 ⫮ (a2 – 3).

Q: For all integers a, b, and c, if a/b and a|c, then a|(5b – 7c). -

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Q: Prove that 1+2+3+4+...+n= n² + n 2 for every positive integer

A: We have to prove the statement using the principle of mathematical induction.We know thatPrincipal…

Q: Prove that Vn e N (the set of natural numbers), ÷+ ɛ N. 3 +

A: We have to use proper manipulation.

Q: |Let A¡ = [i − 1, i] for all integers i = 1,2,3,4. Find the following. a. A₁. A₂- b. A₁ U A₂ U A3 U…

A: The given set Ai=i-1, i for all integers i=1, 2, 3, 4. We have to find, a) A1, A2, A3, A4 b)…

Q: Show that there exist no two positive integers nandr(r<n)such that "C, "C,+1 "C+2 and "C,.3 are in…

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Q: what is (a'+b,a+b) Wher e a and b are relatively prime integers that are not both Zero1

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

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Q: Let i be a positive integer and A; =] -i+1,1[. Then N,A;= O 10,1] None of these O 10,1[

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Q: Show that if a1,: a2., an are integers that are not all 0 and c is a positive integers, then…

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Q: Let i be a positive integer and A; =[-i+1,1]. Then U A;= %3D O 1-0,1[ O [-1,1] O R O 1-0,1] O None…

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Q: Prove that 8 | (54" +7) for every positive integer n.

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Q: 6: Prove by contradiction that for any positive integer x, x+ 49 x ≥ 14.

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Q: Prove that 3 divides n³ +2n whenever n is a positive integer.

A: Introduction: Consider a starting point for which the assertion is correct. The proposition is true…

Q: Let i be a positive integer and A, = [-i+1,1]. Then U, A= O [-1,1] O None of these O l-00,1[ O…

A: Given that          for i be a positive integer…

Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

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Q: If a is any integer, is a(a + 1) even or odd?

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Q: Prove that 1.3 + 5.7 + 9.11 + =

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Q: Prove or disapprove that if a|bc, where a,b,c are positive integers and a≠0, then a|b or a|c

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Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

Q: Let i be a positive integer and A, = {[.., i – 6, i – 4, i – 2, i, i + 2,i + 4,1+ 6, . ). Then N 4…

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Q: Prove that 4 does not divide n2 +2 for any integer n.

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Q: Prove: Any positive integer can be represented as an aggregate of different powers of 3, the terms…

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Q: Evaluate. 13!/12! A.6 B.26 C.12 D.13

A: Recall that, n!=n(n-1)!

Q: Let i be a positive integer and A; =] – i + 1,1 [. Then N, A;= O [0,1[ Ø O 10,1[ O 10,1] None of…

A: Intersection of sets which are forming ascending chain(i.e. A1 contained in A2, A2 contained in A3,…