Prove: If x is a positive prime integer, then x+7 is a composite integer. (i.e., not prime)

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 the bi-conditional: prove that if 5n+2 is even, then n is even, where n is an integer.

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Q: Prove that if n is an integer and n3 + 5 is odd, then n is even

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Q: Let n be a positive integer. Prove that 1:(1)+2 (;) . (:)- n n + 2 . +n . n2"-1. ... n

A: We have to prove the identity: 1·n1+2·n2+....+n·nn=n2n-1

Q: Prove by mathematical induction that if n is any positive odd integer, then 1 + 3n is divisible by…

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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: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

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Q: Prove 1 +,+ i++ = 2 – for all positive integers n. |

A: We want to prove 1 + 12 + 14 + ... + 12n = 2 - 12n

Q: Let p be a prime number and a be a positive integer. Prove that p | (a^p - (p-2)!a).

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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 if n is an integer and 5n + 2 is even, then n is even using a proof by contraposition. a…

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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: 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: Number theory.

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

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

Q: Prove that 8 | (54" +7) for every positive integer n.

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

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

A: Given,  2+7+12+...+(5n-3)=12n(5n-1)

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

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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.…

Q: How to solve this?

A: According to the given information, let p be a prime number such that p>2. It is required to…

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: (b) Prove that there are infinitely many prime numbers of the form 6n + 5

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Q: Let n be a positive integer. Prove that if n2 is even, so is n.

A: We have,  n be a positive integer and an even number. Therefore, n=2k; k∈ℝ

Q: Prove by induction that (2n+1)3 is an odd number for all positive integers n.

A: First principle mathematically induction : Suppose is a mathematical statement .Here n is a natural…

Q: Show that if n is a positive integer, then (n + 1, n2 - n + 1) = 1 or 3

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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: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

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

A: (if part) : Let, n be an even positive integer. Then, n = 2k for some positive integer k. So, 7n+4 =…

Q: . Let n be an integer. Prove that n is odd if and only if 3n? – 7 is even.

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Q: 1. Let n be an integer. Prove that if n is even, then n2 + 2n + 3 is odd.

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Q: Prove that a positive integer n is p

A: Topic :- Number system

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 it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

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Q: Prove, using any method, the following statement: if n is an integer, then (3n+1)(5n+4) is an even…

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Q: Prove that if n even, then n is an integer and 3n+2 is even using is a) A proof by contraposition.…

A: Please refer to the image below

Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

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: 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: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: - Prove that if n is an integer, then n = +. (Hint: use the proof by cases with respect to even and…

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Q: Let n > 1 be an integer. Show that if pn for all primes p≤ √n, then n is either prime or the product…

A: Introduction: Proof by contradiction, sometimes referred to as indirect proving or the reductio ad…

Q: 3.3. Proof that for all integers numbers n, if n3 is odd then n is odd, using proof by…

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Q: Let n be an integer. Show that if n 2 is a multiple of 3, then n must also be a multiple of 3. (a)…

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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: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

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Statements and Logic

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