18 Show that if p is a positive integer such that both p and p² + 2 are prime, then p= 3.

Q: 1. If p>9 2 5 and p and q are both primes, prove that 24 p - q.

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Q: Let a,b be integers, not both 0. Show { na + mb : n,m ​​​​​are integers} = iff a and b are…

A: We have to prove the given statement.

Q: - 2017 q3 – 345

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Q: Prove that pqr is a deficient number, where p, q, r are three distinct primes with p <q < r

A: We have to prove that pqr is deficient number where p, q and r dustinct primes  Such that p &lt;  q…

Q: Determine which of the following “proofs” are correct and which are incorrect. If a proof is…

A: To determine which of the proof are correct and which are incorrect and also it's type .

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: ) Give the contrapositive of the following statement concerning a positive integer p: If p > 2 and p…

A: Contrapositive of a conditional statement of form  (p^q)→r is  ~r→(~pv~q)

Q: Determine which of the following “proofs” are correct and which are incorrect. If a proof is…

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

A: See solution below

Q: 6. Prove that n, n+2, n +4 are all primes if and only if n = 3.

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Q: For any integer ? show that the pair of integers 7? + 9 and 4? + 5 are relatively prime.

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Q: Prove there are no natural numbers n and p with n ≥ 2 and p² + 1 = 2”.

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Q: Show that for every positive

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Q: Show that if p is a prime number, a and b are positive integers and p | ab, then p | a or p | b.

A: The objective is to prove that if p is a prime number, a and b are positive integers and p | ab,…

Q: Let p be a prime number. Prove that (p -1)! a power p + (3p power2  +1)a is divisible by p, for any…

A: Consider the provided question, (a)

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: Let n = pp2 Z+ where p₁ and p2 are distinct prime numbers and T1, T21. If o(n) = 60, then determine…

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Q: Suppose p and q are odd primes such that p = q + 8a for some positive integer a. () = () = ( ) .…

A: Legendre symbol:  For any prime number p and any integer a that is not divisible by p, the Legendre…

Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: Show that if p = q + 4a (p and q are odd primes), then (p/q) = (a/q).

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Q: prove that there exists a prime number p such that n <p< 2n for n = 10,11,12,13,14, 15, 16, 20.

A: for n=10 then take p=19, 10&lt;=p&lt;=20  for n=11 then take p=19, 11&lt;=p&lt;=22 for n=12 then…

Q: Let p be a prime number with p>=5. Prove that p must either be one more or one less than a multiple…

A: We use a fact that any number greater that 5 must be written as one of the form…

Q: p, q, pq | (apq − ap − aq  + a) for all a.

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Q: Let i be a positive integer and A, = {[.., i – 6, i – 4, i – 2, i, i + 2,i + 4,1+ 6, . ). Then N 4…

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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: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

Q: Prove that there exists a positive integer n >1 such that 2n2 + 3n + 11 is prime.

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Q: Prove that a positive integer n is p

A: Topic :- Number system

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 that if p> 3 is a prime number, then 24 | (p2 - 1).

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