A perfect cube is an integer x such that there exists a an integer y and x = y³. Prove that the sum of 3 consecutive perfect cubes is divisible by 9. For example, 13 + 23 + 3³ = 1+8+27 is divisible by 9.

Q: Let x and y be positive integers such that the prime factorization of x has exactly five 3's and the…

A: x = 35×k          k is an integery = 33×l           l is an integer

Q: If x is any positive integer , then the sum of 6x and 11x is always divisible by what?

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

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: Prove that the product of any k consecutive positive integers is an integer multiple of k!.

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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: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: Prove by mathematical induction: for all integers n, if a 2n × 2n checkerboard with alternating…

A: Let Cn be a 2n×2n checkerboard with alternating black and white squares. We have to prove that if…

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

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Q: Prove that for all integers a, b, and c if a divides b then for all naturla numbers n a^n divides…

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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: Find all integers n such that (n^2) - 1 is prime.

A: n² - 1 = (n-1)(n+1)  Now, if (n² - 1) needs to be a prime, then, either n-1 is equal to 1 or, n+1 is…

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

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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: 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: 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 if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

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Q: In the 1989, Journal of Number Theory paper, N. Tzanakis and B. de Wegner showed that the largest of…

A: We have to factories the equation  x3+3x2+2x-258,474,216=0

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: Show that for every positive

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

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Q: Let k be some positive integer. Prove that a number of the form 4k +3 can never be the sum of two…

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

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Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

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

A: We need to prove there are infinitely many integers n such that 4n^2+1 issimultaneously divisible by…

Q: n + 5 > 4 n

A: We know, the square of an integer is always non-negative (n - 2)² ≥ 0 for all integers n

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

A: Given the data

Q: Prove that if a > 0, k > 1, and ak -1 is prime, then a = 2 and k is prime. %3D

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

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Q: C) Prove that the product of k consecutive positive integers is divisible by k!

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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: Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 +…

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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 by contradiction, with example problem: "For any integer n, if n2 is divisible by 4, then n is…

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

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Q: 74. P is a natural number of at least 6 digits and its left- most digit is 7. When this leftmost…

A: Introductions :

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

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Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.