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

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 if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

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.

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

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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: 5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.

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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: Prove that if n is an odd integer then 7n+5 is even.

A: Topic:- algebra  Note :- if more than one question uploaded than only first question need to be…

Q: Prove or disprove: For every positive integer n, 22n – 1 is divisible by 3. -

A: I have proved the result using mathematical induction

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

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

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

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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: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

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

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

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

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,  2+7+12+...+(5n-3)=12n(5n-1)

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 if n is an integer and 5n + 6 is even, then n is even.

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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: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

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

A: Consider the provided expression,

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: 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 the statement is true for every positive integer n.

A: We have, Pn=a2n-1+b2n-1 Let us consider the statement Pn is true for ∀n∈ℕ. Therefore,…

Q: Show that there are infinitely many integers n such that 4(n^2)+1 is simultaneously divisible by 13…

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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 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: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

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

A:

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: Show that for any integer n ≥ 2, C(2n,n) > C(2n,n-1)

A:

Q: 2. Prove that if 2n² + 7n is even and n is an integer, then n is even.

A: We have to prove that the given statement is true by the help of mathematical induction

Q: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

Q: 5. Prove that for any integer n, at least one of the integers n, n+2, n +4 is divisible by 3.

A: According to guidelines we can solved only ist question thank you   we have to Prove that for any…

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

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Q: Prove that if n is an integer, then 2n2 + n + 1 is not divisible by

A: The given problem is to prove the following statement using the possibile cases. We need to find the…

Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

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Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A:

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