If p and q are twin primes, then which of the following is true. i. p + q is divisible by 3 ii. pq = n2-1 for some integern (p- q)2-1 = 3 iii. iv. pq is always an even number. (a) i, i (b) i, ii, iii (c) i, iii, iv (d) ii, iii, iv (e) All of these

Q: For which odd primes p is 3 a square mod p?

A:

Q: 2. Euclidean Algorithm Use the Euclidean algorithm to find gcd(12345, 678).

A:

Q: * For each integer n, let Cn = {n²,n³, nª}. How many elements are in each of C-2, C-1, Co, and C3?…

A: The set always contains the distinct elements. Given Cn = n2 , n3 , n4 Consider C-2 = (-2)2 , (-2)3…

Q: There are infinitely many primes of the form 6n+5.

A:

Q: For a natural number n, we define n! = 1·2·3· ... · (n − 1) · n. For example: 2! = 1 2,3! = 1 2 3 =…

A:

Q: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

A:

Q: 3 Р 1 ? What if 3 is replaced by = 2. For which primes p is -3, 5, -5, 6, or -6 ? Any patterns? Is…

A: The symbol  is the Legendre symbol, used in number theory to denote whether a number is a quadratic…

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

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: 7. Prove that n, n +4, n +8 are all primes if and only if n = 3.

A:

Q: The sum of the first n natural numbers, S = 1 + 2 + 3 + first n natural numbers is 171, determine…

A: Consider the given information:

Q: .If p 5 is an odd prime, prove that either p? – 1 or p² + 1 is divisible by 10.

A:

Q: Show that is composite for all primes p. p+14p²-8

A: The composite primes are those numbers which are generated by multiplying the two smallest positive…

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: (3) If today is Thursday what day of the week will it be 31 days from today? (4) If it is currently…

A:

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

A:

Q: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

A:

Q: let n be an integer solution to the equation n2+an-b=0 (n>7) if the representation of a in base n is…

A:

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

A:

Statements and Logic

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