Prove by cases that n2−1 is divisible by 3 when n is an integer not divisible by 3.

Q: Prove by cases that 25x² + 35x is an even integer whenever 5x-7 is an integer. Note here that the…

A: Apply the condition for 5x-7 to be an integer case by case.

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 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: n^2 +1 is not divisible by 3 for every integer n.

A: Any integer n can be written in the form of: n=3a+b, where b = 0 or 1 or 2 Case 1: If b=0, n2+1=…

Q: Show that for any integer n, n² – 2 is not divisible by 3.

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

Q: Prove that vn if n is divisible by 3 then n2+2 is not divisible by 3

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Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

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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: For every positive integer n, prove that 7n – 3n is divisible by 4.

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

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Q: Show that an integer N is divisible by 9 if and only if the sum of its digits is divisible by 9

A: Let a=anan-1...a3a2a1 be an integer and S be the sum of the digits. Consider a,…

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

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

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Q: Show that for any integer n , n2 – 2 is not divisible by 3.

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Q: Prove: For any integer n, if n2 is divisible by 4, then n is even

A: I'm proving this using by contradiction.

Q: Prove that 2^10 + 5^12 is a composite number.

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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: Prove that for any n E N, nº + 2n is divisible by 3.

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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: Prove that for all positive integers n, 4n-1 is divisible by 3.

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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: イの Prove by mathematical induction that 3" + 7 is divisible by 8 for all integers n > 0

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Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

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

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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: Prove that there is no integer n for such that 5n+3 is divisible by 5.

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

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Q: Prove that2.7n + 3.5n – 5 is divisible by 24, for all n ∈ N.

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Q: Find an integer that leaves a remainder of 2 whan divided by either 3 or 5, but that is divisible by…

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Q: Show that if n is an odd integer then []

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Q: Prove that n3+(n+1)3+(n+2)3 is divisible by 9 for all non-negative integers  n.

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

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Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

Q: Prove that Vn if n is divisible by 3 then n2+2 is not divisible by 3

A: If a and b are two integers. Suppose n is non zero integer such that |n|≠1. If a is divisible by n…

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

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Q: Prove that n2 + 4 is not divisible by 3 for all integers n.

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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: Show that 3-4" + 51 is divisible by 3 and 9 for all positive integers n.

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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: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

Q: Prove that 3" + 1 is divisible by 4 whenever n is an odd positive integer.

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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: Prove that for a positive integer n we have that Bn+3 - 26n – 27 is divisible by 169.

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

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Q: Prove that any positive integer N is divisible by 11 if and only if the difference between the sum…

A: Consider the provided question, we need to prove that, Any positive integer N is divisible by 11 if…

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

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Q: divisible

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