Let i be a positive integer and A, = {..., i – 6, i – 4, i – 2, i, i + 2, i + 4, i + 6, ... }. Then N, A= %3D

Q: Let n denote an integer. Prove that 5|n and 7|n if and only if 35|n.

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Q: Prove that if n is an odd integer, then n3 +n+1 is also an odd integer.

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Q: Prove that any positive integer greater than or equal to 8 can be written as a sum of 3's and 5's.…

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Q: Let i be a positive integer and A =] - i+1,1]. Then N1 A O 10,1[ O 10,11 O 10,1] O None of these

A: ]0, 1] is correct

Q: Prove that any positive integer greater than or equal to 8 can be written as a sum of 3's and 5's.…

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Q: Let i be a positive integer and A, = [-i+1,1]. Then U, A O R O 1-0,1] None of these -1,1]

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Q: Let a be a positive integer. Then Lcm(2a+1,2a+2)= *

A: The given numbers are 2a+1 and 2a+2. Clearly, the given numbers are two consecutive numbers.

Q: Determine for which positive integer values of n, 3n3 +2<n* and prove your claim by mathematical…

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Q: Let $n$ be a positive integer. Among C(2n,0),C(2n,1),..., C(2n,2n), C(2n,n) is the largest. True…

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Q: Prove, for all positive integers n, 1 3 2n – 1 1 - 2 4 2n (3n

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Q: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

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: For every positive integer a, find a composite number n such that n|a^n - a.

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Q: 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[

A: Ai = ] -i+1, 1 [  A1 = ] 0, 1 [ A2 = ] -1, 1 [ Thus ∩Ai = ]0, 1[

Q: Prove that o – 1 is divisible by 4.

A: Even numbers are numbers divisible by 2. Odd numbers are numbers that are not divisible by 2. The…

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

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

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Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

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Q: 3. Prove that if a > 1, then a"/n → o.

A: We used binomial inequality

Q: 1. Let, a1 = 7 and for n > 2, an = "1 express an in terms of n.

A: According to our guidelines, we can solve only the first question. Please repost other questions as…

Q: Let i be a positive integer and A, = {.., i – 6, i – 4, i – 2, i,i+2,i+4,i+6,.. ). Then NA= O set of…

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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: How many 2-permutations are there of a set of eight objects? Prove that for all integers n > 2, P(n…

A: The formula of permutations is given as nPr = P(n, r) = n!/(n-r)!

Q: (a) Show that if x E Q is such that |x|, < 1 for all prime numbers p then x E Z.

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Q: Let n be a positive integer. Prove that if n2 is even, so is n.

A: We have,  n be a positive integer and an even number. Therefore, n=2k; k∈ℝ

Q: prove that there exists a prime number p such that n <p< 2n for n = 10,11,12,13,14, 15, 16, 20.

A: for n=10 then take p=19, 10&lt;=p&lt;=20  for n=11 then take p=19, 11&lt;=p&lt;=22 for n=12 then…

Q: Let i be a positive integer and A; = {...,i – 6, i – 4, i – 2, i, i + 2, i + 4, i + 6, ... }. Then…

A: Just substitute i equal to 1,2,3,... And you'll observe a pattern.

Q: Show that if n is a positive integer, then (n + 1, n2 - n + 1) = 1 or 3

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Q: Suppose that P(A;) = 1 for all i = 1, 2, .... Show that P(N A;) = 1.

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Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

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Q: Let n be a positive integer. n+1 n+1-r (a) Suppose r is an integer amongst 0,1, - n. Prove that…

A: as you have specifically mention only first part i.e. 2)a) here is the solution.

Q: Let i be a positive integer and A, =1 -i+1,1]. Then A- O 10,11 10.1] O 10.11 ONone of these

A: solution of the given problem is below..

Q: If a is an integer such that: 2 ła, 5ła Show that (a)⁴ⁿ = 1(mod 40) for any positive integer n

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Q: 15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

A: Given The given expression is 11·5+15·9+·  ·  ·+14n-34n+1=n4n+1

Q: Let i be a positive integer and A; = {..., i – 6, i – 4, i – 2, i, i + 2, i + 4, i + 6, ... }. Then…

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Q: Let i be a positive integer and A, = {..., i – 6, i - 4, i – 2, i, i + 2, i+ 4, i + 6, ... }. Then…

A: First option is correct

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,…

Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: 11. If n is an odd integer, show that n + 4n² + 11 is of the form 16k.

A: If n is an an odd integer then we have to show that n4 + 4n2 + 11 is of the form 16k for some…

Q: Let i be a positive integer and A; = [1,i+1[. Then U A OR O [-1,1] O [1, [ O 11, [ None of these

A: [1,inf[ is correct answer

Q: Let n > 1 be an integer. Show that if pn for all primes p≤ √n, then n is either prime or the product…

A: Introduction: Proof by contradiction, sometimes referred to as indirect proving or the reductio ad…

Q: Let n>=2 and k be any integer, then prove that (n-1)^2 divides (n^k )-1 iff n-1 divides k.

A: See the attachment

Q: Prove by direct method: for any n integer,if n is odd then 3n+2 is odd

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

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Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

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Q: is a prime number and that n is an integer such that p|n^2 prove that it follows p|n

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Q: Let i be a positive integer and A; =[1,i+1[. Then U A;= O [-1,1] O R O 11, [ O [1, [ None of these

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Q: Let a and b be positive integers such that ax + by %3D 7 for some integers x and y. Then O…

A: Greatest common divisior:  Let a and b be two integers such that gcd(a,b)=d if  d|a and d|b If…