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

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: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

Q: Prove that the statement is true for every positive integer n.

A: Given, 3+9+15+..+(6n-3)=3n2

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 the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n.

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: Prove that 5 does not divide 5n+3 for any integer n.

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Q: Q2 Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

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Q: Prove that the statement is true for every positive integer n.

A: To prove: 1+4+7+...+3n-2=n3n-12

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 the statement is true for every positive integer n.

A: Given that, 2 is a factor of n2+n   We have to prove this statement is true for every positive…

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 every positive integer n, 2"> n

A: As per the guidelines, since there are more than one question only one question will be answered.…

Q: Find all integers x + 3 such that x – 3 is divisible by x – 3. |

A: To find all integers such that x^3-3 is divisible by x-3

Q: Show for all integers n> 1 that 7 I (3ªn+l – 5²n-1 ).

A: a) Prove 734n+1-52n-1 for all integers n≥1 by mathematical induction as follows. Basis step: For…

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

A: answer

Q: Prove that the statement is true for every positive integer n.

A: Given arithmetic progression (AP) is: 2+5+8+.........+3n-1 Here the first term is 2 and the common…

Q: Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least…

A: Here we have to prove that if for some integers a,b,c we have that a3+b3+c3 is divisible by 9,then…

Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n.…

Q: Find all integers n such that n| (2n+3). *

A: Given, n|(2n+3)     TO find, All integers n

Q: Prove that the statement is true for every positive integer n.

A: Given,  2+7+12+...+(5n-3)=12n(5n-1)

Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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

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Q: Prove that the statement is true for every positive integer n.

A: we have to prove that the statement is true for every positive integer n. 4 is a factor of 5n-1

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

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Q: Prove that the statement is true for every positive integer n.3 is a factor of n3 + 2n

A: Consider the given statement, "3 is a factor of  n3+2n" for all positive integer n Suppose Pn…

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

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Q: Prove that I +:+ + for every positive integer n.

A: In this question, principle of mathematical induction is used. Principle of Mathematical Induction…

Q: Find the smallest integer i for which the statement (1 + i2 )<2' is true. Then, prove the statement…

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

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Q: Prove that the statements here are true for every positive integer n. a + (a + d) + (a + 2d) + · · ·…

A: To Determine: Prove that the statements here are true for every positive integer n. Given: we have…

Q: For every integer n23 I+ 2+ 2?+ +4^= 4(4n-16) %3D 3. Proof (by induction)

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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: For every positive integer a, find a composite number n such that n|a" – a.

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Q: Prove that the statement is true for every positive integer n.

A: Given, Prove this statement is true for every positive integer n: 2+4+6+..+2n=n(n+1)

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

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Q: Show for all integers n2 0 that 91 (4" - 1). a.

A: A. We will solve the question using the method of mathematical induction.

Q: Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

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

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Q: Prove that the statement is true for every positive integer n.

A: a&gt;1, then an&gt;1

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