Prove that for any odd positive integer, n, the sum of nconsecutive positive integers is divisible by n.

Q: Prove that any positive integer greater than or equal to 8 can be written as a sum of 3's and 5's.…

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Q: ) Prove that Prove by induction that 11" - 6 is divisible by 5 for every positive integer n.

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Q: Prove that 6 divides n^3 − n whenever n is a non-negative integer

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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 for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: Prove that if n is a +ive integer,

A: Given that, M xddxnf(x);p = (-1)npnF(p) We can prove this by mathematical induction.

Q: 1. Prove that nº + n is even for every integer n.

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Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: Prove that the statement is true for every positive integer n.n < 2n

A: n<2n Let us prove this statement by using the Principle of Mathematical Induction. When n=1,…

Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x > – 1.

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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: Prove by contradiction : For any odd integer n : n² – 7 is not divisible by 4.

A: Proving a statement by contradiction means assume the negation of statement true and prove the…

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: Number theory.

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Q: Prove that the sum of any five consecutive positive integers is always a multiple of five.

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Q: Prove that for all natural numbers n, (n^3/3) + (n^5/5) + (7n/15) is an integer

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Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

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Q: Prove that for any positive integers m and n there are m consecutive positive intege numbers such…

A: As per the question we have to prove that for any positive integers m, n there exist m consecutive…

Q: n any positive integer n, there exist n consecutive composite integers

A: Given that given any positive integer n, show that there exist n consecutive composite integers.

Q: 7) Prove that for any integer n, n' -n is even.

A: We need to prove that for any integer n, n3-n is even, Now, an integer can be either even or odd.…

Q: Define the "sum" of one integer to be that integer and use strong mathematical induction to prove…

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Q: Prove that there is some positive integer n such that n, n + 1,n + 2,... , n + 200 are all…

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Q: Let X = {x;} be a set of n positive integers. Show that the sum of the integers of a subset of X is…

A: It is required to show that sum of elements of a subset of described set, X, is divisible by integer…

Q: Prove that if n is an integer and n² is an even integer, then n is an even integer

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Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: Given 51 distinct positive integers Strictly of Prove two trom to Sum 99 less than 100,

A: Please find the answer in next step

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

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

Q: Let n be an integer. Prove that if 7 divides (n – 2) or 7 divides (n – 5), then 7 divides (n² + 3).

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Q: Prove that for any integer n, n' -n is even.

A: CASE I When n is even Let n=2m,m∈N⇒n3−n=2m3−2m⇒2m4m2−1=2η,where η=m4m2−1Thus,n3−n is even when n is…

Q: Prove that for every positive integer n, there are n consecutive composite integers.

A: This question is related to Descrete Mathematics

Q: Prove that N ≈ O*, where N is the set of natural numbers, and O* is the set of positive odd…

A: To prove:

Q: Prove that if n, k are positive integers with k | (n +4) and k| (n – 1), then either k = 1 or k = 5.…

A: k | (n+4)  k | (n-1)  => k | {(n+4) - (n-1)} => k | 5

Q: Prove that 3 divides n3+2n whenever n is a positive integer

A: We use mathematical induction. Let P(n) be "3 divides n3+2n" For n=1 n3+2n=(1)3+2(1)=3 We then note…

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 a positive integer n is p

A: Topic :- Number system

Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

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 any integer greater than or equal to 48 can be represented as the sum of nonnegative…

A: Principle of mathematical induction: For each and every natural number n, mathematical induction is…

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

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Q: Prove 8^ +l is composic for any poSitive integer n.

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Q: Prove that 30 divides n° – n for every nonnegative integer n. -

A: The given question can be solved as shown in step2.

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

A: Solution

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 for any integer n, if n³ is odd then n is odd.

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Q: Prove that the sum of two consecutive odd integers is divisible by 4.

A: Let the two consecutive integers be n and n+2 Then we need to prove that (n)+(n+2) is divisible by 4…

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: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…