Prove for every integer n > 8 that there exist nonnegative integers a and b such that n = 3a + 5b.

Q: Solve the following oblique triangle using the Law of Sines. Show your work! Round 1 place after the…

A: We are provided with following information : B=47.2o , C=73.7o b=56.9 ft According to Sine rule , we…

Q: (12) Let p be a prime integer, p≥ 5. Prove that p² - 1 is divisible by 24.

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Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: 3n+2 and 5n+3 are relatively prime for every integer n.

Q: Compute for an arbitrary integer k

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Q: Prove that every positive integer n>1 has a prime factorization.

A: Prime factorization is a way of expressing a number as a product of its prime factors.

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

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Q: prove that for all integers m, n & d if d/m & d/n then d/(2m+3n)

A: We know that if a and b are integers such that a/b then b is a multiple of a. i.e. b=ka where k∈ Z{…

Q: Prove that if n is an even integer greater than 2, then 2 - 1 is a composite.

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Q: Prove that if n is an even integer greater than 2, then 2 - 1 is a composite.

A: Answer

Q: Prove that for any natural number N, there exists N consecutive integers none of which is a power of…

A: Please see the attachment

Q: An integer is square-free if it is a product of distinct primes. Show that for each integer k≥ 1…

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Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

Q: prove that n(13n²-1) is divisible by 6 for every n ≥1

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Q: Prove that for any odd positive integer, n, the sum of nconsecutive positive integers is divisible…

A: To prove for any odd positive integer, n, the sum of nconsecutive positive integers is divisible by…

Q: Let b a natural number and q1, q2, r1, r2 integers with 0 ≤ r1 < b and 0 ≤ r2 < b such that q1b + r1…

A: Natural Numbers: Natural numbers, that embody all positive integers from one to time, square measure…

Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

Q: Prove that 3 devides 17k+1-2*5k+3+11k+3whenever k is a non-negative integer.

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Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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Q: Show that for every positive

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Q: A $ 100 deposit is made into a college fund at the beginning of each month for 10 years. The fund…

A: Consider the given information. The deposit amount is $100. The rate is 9% and the time is 10 years.…

Q: Prove that the integers n, n+2, n+4 are all primes if and only if n=3

A: Statement: Prove that the integers n, n+2, n+4 are all primes if and only if n=3 Proof by…

Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: prove that in any eight composite positive integers not exceeding 360, at least two are not…

A: Prove that in any eight composite positive numbers not exceeding 360, at least two are not…

Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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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: Prove that n!+1 and (n+1)!+1 integer n. relatively prime for every positive

A: Given, n!+1 and n+1!+1are two numbers and to prove that they are relatively prime.

Q: Select the following statements that are true. gcd (2022, 111) = 1 2023 is a prime gcd (2022, 1685)…

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Q: Prove that a positive integer n is prime number.if no prime p less than or equal to n divides n.

A: Let us assume that the contradiction of the statement , let  n ≥ 0 be a composite number .  Thus , n…

Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

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Q: Prove that for every ponitive integer a if aº – 6° is a prime for some positive integer 6 5 = 2 or 5…

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Q: Prove that ifnis an integer and3n+ 2is even, thennis even using a proof by contraposition.

A: Contra-positive: "If q then p" is the contra-positive of a conditional statement of the type "If p…