4) Prove that the product of any three consecutive integers is divisible by 6.

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: What is the converse of: an integer has divisors other than 1 and itself, if an integer is…

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Q: Prove the statement. "For any integer n, nº +4 is not divisible by 8." by Contradiction.

A: Given: n be any integer. To prove: n3+4 is not divisible by 8.

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: A natural number greater than 1 that has only itself and 1 as factors is called a/an number.

A: Given that  A natural number greater than 1 that has only itself and 1 as factors is called…

Q: 2. Prove that the product of any two odd integers is an odd integer.

A: Let x and y be two odd numbers. If a and b are odd, then a and b can be written as: x=2m+1 , y=2n+1…

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: a

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Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

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Q: Show that every positive integer can be written as the product of a square (possibly 1) and a…

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

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Q: Prove by using direct proof: Prove that the sum of any three consecutive integers is divisible…

A: Solution   Prove that the sum of any three consecutive integers is divisible by 3.

Q: Show that if m is an integer then 3 divides m3−m.

A:

Q: Show that if any eight positive integers are chosen, two of them will have the same remainder when…

A: Pigeonhole Principle- If n items are assigned to m containers, and m<n,then at least one…

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: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

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

A: The detailed solution is given below Thank you

Q: 1- Show that if a is an integer, then 3 divides a^3-a. with details plz

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Q: what two consective odd integers have a product of 63?

A: Find two odd integers.

Q: Suppose a and b are integers that divide the integer c. If a and b arerelatively prime, show that ab…

A: Since a/c   &  b/c  , we know there exists integers K1 & K2  ,with K1 a=K2 b=c--1Also…

Q: 7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is…

A: Using binomial theorem:  n0+2n1+22n2....2nnn=n01n·20+n11n-1·21+n21n-2·22....nn1n-n·2n…

Q: (b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

A:

Q: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

A: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

Q: (a) Suppose that there are k distinct positive integers that divide an odd positive integer n.…

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

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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: Prove that if the sum of the digits is subtracted from a natural number, then the result is…

A: let us assume the number to be three digit number xyz therefore, xyz=100x+10y+z sum of digits will…

Q: Prove that 11 divides 3* 5k + 42k+1 + 33k+2 – 5 * 72k for every k non-negative integer

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

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

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

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Q: Prove that it is impossible to choose three distinct integers (a – b)|(b – c), (b – c)|(c – a), (c –…

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

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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: 4. Prove that if m and n are positive integers and x is a real number, then

A: We have to prove here that the expression here follows both commutative and associative property. If…

Q: Show that every odd integer is the difference between two consecutive perfect squares.

A:

Q: that the set of negative integers divisible by 3 is countable

A: Set of negetivee integers divisible by 3 isS=-3n, n∈ℕf: ℕ→S   defined asf(n) = -3n,To check…

Q: Show that in any collection of 7 or more natural numbers there are ALWAYS TWO numbers whose sum or…

A: In the given problem we have to show that in any collection of 7 or more natural numbers there are…

Q: Let suppose integers are randomly selected. Prove that the difference of two of them is divisible by…

A: Solution

Q: Prove that if the product of two integers ab is odd, then a2 + b2 is even

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Q: a2 − b2is divisible by 5.

A: First observe that one's place of the square of a positive number can only one among 0,1,4,9,6, and…

Q: A restaurant sell features a burrito bar in which a customergets to choose one of 2 different kinds…

A: As per bartleby guidelines only one question is answered please upload the other one separately…

Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

A:

Q: divisible

A: