Show that the the greatest common divisor gcd(16" – 4",17" – 2") is divisible by 3 for all n E N.

Q: Let n ∈ Z. Prove that if n is divisible by 3, then n = 6k or n = 6k + 3 for some integer k.

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Q: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

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Q: Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

A: n is an integer. To show: 2n3 + 4 is divisible by 4 then n is even.

Q: Prove that for any integer n, n6k - 1 is divisible by 7 if gcd(n, 7) = 1 and k is a positive…

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Q: 1. Prove that 7" -1 is divisible by 6 for all n > 1.

A: As per Bartleby Guidelines we can solve only one question at a time. Hope you understand

Q: For any a ∈ N, prove that a3 − a is divisible by 3. [Hint: Split into three cases: a = 3n, a = 3n +…

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Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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Q: Let n be an integer. Prove that if 2n +4 is divisible by 4, then n is even.

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Q: Prove that 5n-1 is divisible by 4 to any positive integer n

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Q: Find all zero divisors in Zn for the following values of n.   .n = 12

A: To Find all zero divisors in ℤ12 Zero Divisor:- Let R be a ring and a ∈R. If a ≠0 and if there…

Q: Prove that for any integer n, we have n - 2 is not divisible by 4.

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Q: Show by induction that 19"-8" is divisible by 11 for all n >0

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Q: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

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Q: prove that n(13n²-1) is divisible by 6 for every n ≥1

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Q: Find all integers n > 1 such that 1n − 1)" is +2+...+(n divisible by n.

A: To find all the integers n>1 such that 1n+2n+...+n-1n is divisible by n

Q: If n is a positive integer prove that 3^(2n) - 26n -1 is divisible by 676

A: if n is a postive integer prove that 3^(2n) - 26n -1 is divisible  by 676

Q: Find all zero divisors in Zn for the following values of n.   n=8

A: We have to find zero divisor in Z8

Q: Use mathematical Induction to prove that 4" + 7" + 10 is divisible by 3 for every non-negative…

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Q: Prove that n3 + 2n is divisible by 3 for all n E N.

A: We prove that n3+2n is divisible by 3 for all natural numbers n.

Q: Prove that n2-n is divisible by 2 for every positive integer n

A: We have to prove that,  n2-n is divisible by 2 for every positive integer n. We will use…

Q: (i) Use the method of Mathematical Induction to show that for all positive integer values n21, (1+…

A: Solution of given problem is followed by

Q: (i) Use the method of Mathematical Induction to show that for all positive integer values n21,…

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

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Q: 2. Prove that n = aŋɑp-1 aza,a, is divisible by 8 if and only if aza,a, is divisible by 8. A…

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Q: For all integers n> 0, 32n – 1 is divisible by 8.

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Q: Prove by mathematical induction that for all integers n 2 0: a 5" + 2 x 11" is divisible by 3, b 33n…

A: (a) Case 1 : n=1 51+2×111 = 5+2×11= 5+22= 27 As 27 is a multiple of 3 , hence the result holds for…

Q: Find all zero divisors in Zn for the following values of n.   n = 10

A: As we know  zn=0, 1, 2, 3, ....., n-1 For n = 10 z10=0, 1, 2, 3, ....., 9 Now use the definition of…

Q: (d) Prove by induction that 5 "– 1 is divisible by 5 for all n e Z +.

A: we have to show that 5n-1 is divisible by 5 for all n belongs to Z+ by induction for n =1.

Q: 1. Let x E Z. Prove that if x4–1 is not divisible by 4, then x2 – 4.x– 3 is also not divisible by 4.

A: As per our guidelines we are supposed to answer only one question.   So, here is the ques. no. 1

Q: Prove that if q^2 is divisible by 3, then q is divisible by 3.

A: To prove that if q2 is divisible by 3, then q is divisible by 3.

Q: Prove that 6-1 is divisible by 5 for any n 21.

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Q: 3. Given that n is a positive integer, prove by induction that, the 7"+ 5 is divisible by 6. Guess a…

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

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: Prove that 9? −1 is divisible by 8,for all n ≥ 1

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Q: If n = 4k + 3, does 8 divide n2 - 1? (Assume that n and k represent integers.) O Yes, because n2 - 1…

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Q: Prove that n² +n is divisible by 2, where n E N.

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Q: Prove that n2+n  is  divisible by 2, where nN.

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Q: 2. Prove by the method of mathematical induction that for any integer n 21 the expression 8" + 13 is…

A: The solution is given as

Q: Prove that for all n E N, 7" +2 is divisible by 3.

A: We will use mathematical induction to prove the result in question 4.

Q: Prove that n2-n is divisible by 2 for every positive integer n.

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Q: Use the PMI to prove that 6" – 1 is divisible by 5 for every integer n > 0.

A: To prove that 6^n-1 is divisible by 5 for every integer greater that or equals to 0.

Q: (a) Prove that 82n – 12" is divisible by 13 for all nɛ N.

A: As per guidelines we are supposed to solve maximum one questions at a time.kindly post another…

Q: If  n×k divisible by 4, then n is divisible by 4 or k is divisible by 4. Which of the following is…

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Q: 5.) Prove that given any integer for n, n³ + 2n will be divisible by 3.

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

Q: Let x be an element of Z. If x is odd, then (x 2+3)(x 2+7) is divisible by 32. (use direct proof)

A: Given that x is an element of Z and it is odd. So x will be in the form 2k+1 for k=0,1,2,..... We…

Q: A. Prove that n3 + 2n is divisible by 3 for all integers n.

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Q: 3. Prove that if p is a prime >3 then the numerator of 1 + 1/2+1/3+...+V/(p-1) is divisible by…

A: Given: Let us consider the given, r=1+12+13+...+1p-1

Q: The greatest common divisor of 9n + 10 and 10n +9 is Oa 10 or 19 оь. 19 O. None d. 1 or 19 Oe. 1

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Q: Prove that n(n + 1)(n + 2) is always divisible by 6

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