Which of the following are twin primes. a) 841 & 843 b) 857 & 859 c) 701 & 703 d) None of these

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

A:

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: Every integer n22 either is a prime or can be expressed of a product of primes. This statement is…

A: Every integer n≥2 is a prime or can be expressed as product of primes. This statement is Fundamental…

Q: If a and b are two distinct prime numbers, then the greatest common divisor of a and b is O ab

A:

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.

A:

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

A:

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

A:

Q: 1 be rati onal.Then, there exist positive co - primes a and b such that a If possible, let b

A: Please see the below picture for detailed solution.

Q: Show that if n is an even perfect number, then n = 2P-'Mp for some Mersenne prime Mp 2P – 1.

A:

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: 24. Determine whether each of these integers is prime, verifying some of Mersenne's claims a) 2 -1…

A: As per guidelines, we will solve first three subparts of the question a) Given integer is 27-1  We…

Q: Prove by using a direct proof that if n is an odd number, then n2 is also odd.

A:

Q: 5-a) How many prime triples of the form (p, 2p + 1, 4p + 1) are there? Describe them all, if…

A: a) The objective is to state the number of prime triplets of the form p,2p+1,4p+1. b) To verify the…

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: 5. Let n be a positive integer and p1,P2,P3, . „Pn be n distinct primes. If m is a positive integer…

A:

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

A:

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

A:

Q: Which positive integer less than 21 are relatively prime to 21? 18 O 19 O 21 O 24 O

A: Which positive interger is less than 21 is relatively prime to 21?

Q: If a, b, c, d are distinct prime numbers with an as smallest prime then a * b * c dis None of the…

A: Given a,b,c,d are distinct prime numbers with a as smallest prime. Then, a*b*c*d=_________

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

A: Prime number is a number p having only two divisors p and 1 only.

Q: Every odd prime number is greater than 2.

A: Given: "Every odd prime number is greater than 2." To find: Statement in the form "If p, then q"…

Q: Show that if n is an integer and 5n + 6 is odd, then n is odd. (Use contradiction.)

A:

Q: If a, b, c, d are distinct prime numbers with an as smallest prime then a *b * c dis None of the…

A: Answer

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: If n is a product of twin primes, then a) Ø(n)o(n) = (n – 1)(n – 3) b) ø(n)o(n) = (n + 1)(n – 3) %3D…

A: If n is a product of twin primes, then a)  ϕ(n)σ(n) = (n+1)(n-3) b)  ϕ(n)σ(n) = (n+1)(n-3) c)…

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: describe (a,p)

A: Given, if p is a prime, and a is a positive integer.

Q: 2. For which odd primes p is 11 a

A: If p is a prime, then by Fermat's Little Theorem

Q: 9. Ifn is a squarc-frce integer, prove that r(n) = 2" , where r is the number of prime divisors of…

A:

Q: Wh of the following are twin primes. 41 & 643 b) 497 & 499 c) 673 & 675 d) None of these

A: Twin prime numbers: Two prime numbers are called twin primes if there is only one composite number…

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

A: Given:

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

A:

Q: All of the following statements about prime numbers are true except

A:

Q: Show that if n is prime and b is a positive integer with n ̸∣ b, then n passes Miller’s test to the…

A:

Q: If n is a perfect number, prove Σ d|n 1 2.

A:

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

A:

Q: If a, bị c, d are distinct prime numbers with an as smallest prime the a * b * c *d is Prime number…

A: Given: a, b, c, d are distinct prime numbers and  a is the smallest prime number We know that, the…

Q: 2. If p > q > 5 and P and q are both primes, prove that 24 p - q² |

A:

Q: 15/If n and n +2 are a pair of twin primes, establish that o (n +2) = 0 (n) +2; this also holds for…

A: σ (n) = sum of all the positive divisors of n

Q: Prove that if p is a prime integer, then 3√p is irrational.

A: To Prove that: If p is a prime integer, then 3√p is irrational.

Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

A:

Q: Wh of the following are twin primes. 41 & 643 b) 497 & 499 c) 673 & 675 d) None of these :Select one…

A: Option (a) is correct

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

A:

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: Give a counterexample to the statement. The product of 2 primes is always odd.

A: Product of two primes.

Q: Prove that for all primes p that if p does not divide n2, then p does not divide n.

A:

Q: O Prove, by induction, that 3" +4" < 5ª for all natural numbers n, greater than or equal to the…

A: (a) The given inequality is: 3n+4n&lt;5n Substitute n=1 into the inequality and check whether the…

Logic and Proofs

Discrete mathematics deals with the separated values of data such as integers. It is widely used in the practical field of computer science and mathematics. By knowing discrete mathematics one can improve their reasoning and problem-solving capabilities. Discrete mathematics became popular after the boom of computers which operate in discrete steps and store data in discrete bits. It also useful in studying computer programming, algorithms, and cryptography.