Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other 8 numbers.

Q: show that if x is an integer then x2+x-41= 0 produce prime numbers

A: Here show that if x is an integer then  produce prime numbers.

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

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

Q: Suppose that the number 'a' has a property that for every natural number n, a ≤ 1/n. Prove that a ≤…

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Q: Prove that if n is an integer and 7n+2 is even, then n is even. State which method you are using for…

A: Let  n be any integer.  We have given that , 7n + 2 is even. We need to prove that ,  n is even. We…

Q: Prove that for every non-negative integer n the value of n² + n+41 is prime ?

A: The number n² + n + 41 is not prime for every natural number n. Let, f(n) = n² + n + 41

Q: Justify that every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q.

A: Let p be a prime number other than 2 and 3. If p is divided by 6, then by the quotient-remainder…

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 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 if n is an integer, then 3n? + n + 14 is even

A: Case I - Let n be an odd integer , then n= 2k+1 where k is an integer. Then 3n2+n+14   =…

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: Prove there are infinitely many prime numbers

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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: Every two different prime numbers are relatively prime. False True

A: In this question we need to check the given statement is true or false

Q: Number theory.

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Q: Prove that there are infinitely many prime numbers

A: Let us assume that there exist finitely many prime numbers (Let n). Here, n is a positive integer.…

Q: Prove: n! > 2"- for any positive integer n that is greater than 2.

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Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

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Q: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

A: Prove or disprove: 2n + 1 is prime for all nonnegative integers n.

Q: 1) Prove that the numbers -197, -239 are prime numbers 2) Find all prime numbers between -90 and 90.

A: number system

Q: Before arriving at his famous conjecture, Goldbach had conjectured that every odd positive integer…

A: Goldbach conjecture says that every integer greater than 5 is the sum of three primes.

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: Suppose p and q are distinct prime numbers. Find the number of positive integers x ≤pq such that…

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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 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: show that if x is an integer then x^2+x-41= 0 produce prime numbers

A: This question is taken from the quadratic equation in which we have to find the roots of the given…

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: Prove that there is a natural number N such that for all natural numbers n > N, 2/n < 0.1.

A: Assume that there does not exist N such that for any natural number n,  n&gt;N, 2/n&lt;0.1.

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

A: Prove that for any integer n, n3-n is even.

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

A: This question is related to Descrete Mathematics

Q: Show that if n is an odd integer then []

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Q: show that no integer of the form n^3+1 is a prime other than 2

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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 the number in between twin primes is always a multiple of 6.

A: A Twin prime are those numbers which are prime and having a difference of two ( 2 ) between the two…

Q: Prove or disprove that for every positive integer n, if n is even then n^{2}+1 is prime.

A: False

Q: Prove that 3, 5, and 7 are the only three consecutive odd integersthat are prime.

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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 every natural number n can be written in exactly one of the following forms: • 3k for…

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Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: Given: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even

Q: Prove by Induction that for every natural number п, 3h-1 is a multiple of 2?

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Q: Show that if 2m +1 is an odd prime, then m 2" for some nonnegative integer n.

A: We solve this question by contradiction

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

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Q: Prove that for every non-negative integer n the value of n^2+n+41 is prime

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Q: Let be a prime number (integer). Show that p2 + 5 cannot be a prime number.

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Q: For any natural number n, n²+1 is always odd. True False

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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: Prove that there is not a finite number of primes.

A: To Prove: There is not a finite number of primes. That is , to prove there are infinite number of…

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

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Q: A man finished a job in 5 days. On the first day, he finished 1/m of the job. On the second day, he…

A: GIVEN THAT JOB FINISHED = 5 DAYS

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.