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

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

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

Q: If n2 4 is an integer,prove by induction that n<nl.

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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: Prove by induction that n3 -n is divisible by 6 for all positive integers

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

A: To show  33n+3 - 26n - 27 is divisible by 169.

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: 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: Prove, by induction, that n + 3n2 + 2n is divisible by 6, for n 2 0.

A: In this problem I have use mathematical induction method two times please read step by step.

Q: (2) Prove by mathematical induction that for all integers n > 0, the number 42n+1 + (21 · 33"+1) is…

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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 if n is a positive integer, then 133 divides 11"+1 + 122n-1

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Q: 4. Prove that given any integer for n, n3 + 2n will be divisible by 3. 5. Write the formal proof…

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

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

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Q: Use induction to prove that for every integer 15"+2 + 162n+1 is divisible by 241. n > 0,

A: To use mathematical induction to prove the given statement

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: Use mathematical induction to prove that 3n+ 7n−2 is divisible by 8 for every non-negative integer…

A: The given statement is, Pn=3n+7n-2 Let us assume that the statement is true for n=1 P1=31+71-2=8 As…

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

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Q: Use Induction to prove that 22n+1 + 32n+1 is divisible by 5 for all n > 0.

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Q: Prove that the fourth power of any integer has one of the forms 5k or5k + 1, where k is an integer.

A: We need to prove that any integer has one of the forms 5k or 5k + 1, where k is an integer.Let p be…

Q: Prove by Mathematical Induction that for any natural number n the number 52n+1 – 22n+1 is divisible…

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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: . Let n be an integer. Prove that n is odd if and only if 3n? – 7 is even.

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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: Find all positive integers n such that the sum of all positive divisors of n is odd.

A: Given the data

Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

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Q: Let n ∈ N and p be a prime. Prove that if n is not divisible by p then n and p arerelatively prime

A: From Prime factorization theorem we know that any natural number can be written as product of…

Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

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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: Use induction to prove that for every integer n  ≥ 0, 5n+2+ 62n+1 is divisible by 31.

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Q: 4. Prove that given any integer for n, n3 + 2n will be divisible by 3.

A: We have to prove by induction.

Q: Prove that 8" – 14n + 27 is divisible by 7, for all positive integers n.

A: The objective is to prove 8^n-14n+27 is divisible by 7.

Q: Prove, using mathematical induction, that 8n−3n is divisible by 5 for all integers n≥0.

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

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Q: 1. Prove: 11" – 6 is divisible by 5 for all positive integers, n.

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Q: Let n be any integer. Using a proof by cases, prove that n4 is either in the form of 5k or 5k+1 for…

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Q: Use induction to prove that 11" – 6 is divisible by 5 for all integers n > 1.

A: First show that given statement is true for n = 1 Then also prove that it is true for any integer…

Q: Use induction to prove that n^3 + 2n is divisible by three for all integers n greater than or equal…

A: Given information:  n3+2n is divisible by three for all integers n greater than or equal to zero.…

Q: For any non negative integer n , prove that an – bn is divisible by a-b,where a and b are any…

A: Use the binomial factorisation