Show that there is a prime number p such that p+4 and p+6 are also prime numbers.

Q: Let p>5 be the prime number. Prove that the expression p4-10p2+9 is divisible by 1920.

A: Let, p>5 be the prime number. Prove that the expression p4-10p2+9 is divisible by 1920.

Q: 20. (a) Show that the sum of two odd prime numbers is never prime. (b) Is (a) true if the word odd…

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Q: A. If p + 2 is a prime number, prove that p? + 3 is divisible by 4.

A: Let p≠2 be a prime number.

Q: Let p > 5 be the prime number. Prove that the expression p – 10p +9 is divisible by 1920.

A: We have to prove that the expression p4-10p2+9 is divisible by 1920.

Q: Want a rigorous proof   Prove that The set of prime numbers has no largest element.

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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: . Show that p(X) = XP-1 +... +X +1 when p is prime.

A: For every integer n≥1, the nth cyclotomic polynomial Φn is the monic polynomial whose complex roots…

Q: p + 2 is a prime number, prove that p2+3 is divisible by 4. (2.5

A: Prime and divisibility

Q: For which prime p,  p^2+2 is also prime?

A: Put p=3 in p2+2 and obtain that, 32+2=9+2=11. Clearly, 11 is a prime number. Thus, p=3 satisfies…

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: All prime numbers are odd.

A: We have state whether given statement is True or False. We know that  A prime number is a number…

Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p, at least one of…

A: GIVEN: Prove that there are no 3 consecutive odd prime numbers

Q: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: Do you thinkthat there are infinitely many primes of the n -1

A: there are infinitely many primes of the n -1 Explain the above statement

Q: How do I prove: If n > 1 is a natural number then there is a prime number p such that p|n.

A: This is a theorem of real analysis in which we have to find that there is a p who divides 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: 4. Find all the prime numbers that can be expressed in the form n2 – 9 for some integer n.

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Q: If n is prime, then 2" – 1 is prime.

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Q: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

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Q: Find (characterize) all primes p > 3 that satisfy (금) 1.

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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 there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of…

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Q: If p>=q>=5 and p and are both primes, prove that 24| p^2-q^2.

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Q: 3. The GCF of any two distinct prime numbers is 1. (Distinct they are not the same number) means

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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: how many pairs of primes p and q are there such that q-p=3? Prove your answer.

A: Here, we need to find all the possible pairs of prime numbers p and q such that the difference (q-p)…

Q: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

Q: Prove or disprove: let n be an integer. if 3 divides n^2, then 3 divides n.

A: Let us consider some cases for proving that n be an integer. if 3 divides n^2, then 3 divides n.For…

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: 18. Numerical evidence makes it plausible that there are infinitely many primes p such that p+50 is…

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Q: Let p>5 be the prime number.Prove that the expression p4-10p2+9 is divisible by 1920.

A: Given: Let p>5 be the prime number

Q: Show that there are no prime triples other than 3, 5, 7.

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Q: Let p be a prime number, n 2 2. Show that 6(pn) = po(n) if and only if p n. %3D

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Q: А. If p 2q 5 is a prime number, prove that p2-q? is divisible by 3.

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Q: Let c = 2^100 and p be the smallest prime such that p >= c. How many composite numbers are between c…

A: Let c=2100 And p=min p is prime:p≥c,p∈ℤ Claim: Total number of composite number between c & p,…

Q: Find all prime numbers pand of for which 2p²q + 45 pp ² = y² 2 де

A: Given:

Q: Suppose that 56 % of students play football, 53 % of students play cricket, and 12 % of students…

A: Let F=students play footballC=students play cricketGiven,P(F)=0.56P(C)=0.53P(F¯∩C¯)=0.12

Q: Prove that 2^(27) - 1 is not a prime number.

A: A prime number is a natural number greater than 1 that is not a product of two smaller natural…

Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 +q2 + r2.

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Q: P(n): 4" – 1 is divisible by 3.

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

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Q: Find all numbers p such that all six numbers p, p+2, p+6, p+8, p+12 and p+14 are primes.

A: For the given statement

Q: How many twin primes are there where both primes are less than or equal to 31? Number of twin…

A: Twin prime: If there is only one composite number between two prime numbers, then that two numbers…

Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 + q2 + r2.

A: We have to prove that 3 divides p2+q2+r2, where p,q and r primes other than 3.

Q: Use proof by cases to show that if n is an integer then n^2 ≥ n.

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Q: Let p ≥ 5 be a prime. Prove that p(p+1) | (p − 1)! +p+1. -

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Q: If p 2 5 is a prime, then p² + 2 1) prime 2) perfect integer 3) composite (Hint: L et n - 6L

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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