5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form 4m +1.

Q: 8. Prove that n!+1 and (n+1)!+1 are integer n. relatively prime for every positive

A: Given, and are two numbers and to prove that they are relatively prime.

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

A:

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

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

A:

Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

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

A:

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

A:

Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

A:

Q: 12.16. Give an example of a prime p of each of the forms: a. 4k + 1 b. 4k + 3

A: Let the prime number be

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: We have to prove that and  are relatively prime for every integer .We know thatThe greatest common…

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: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

Q: 3. Prove that 2n > n² for all integers n such that n > 4.

A:

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

A:

Q: 10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime.

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Q: Find the smallest positive integer N≥ 2 that gives remainder 1 upon division by each of the integers…

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

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Q: Show that 32n 1 is always a multiple of 8 for all n = N = {1, 2, 3,...}. We say one integer A is a…

A: Step 1:Step 2:Shows a proof by induction that 32n - 1 is always a multiple of 8 for all natural…

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

A:

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: 2. Prove that if n is even, then 3n + 1 is odd

A: The given expression is 3n + 1. Prove that 3n + 1 is odd, if n is even.

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

A:

Q: Prove that 31 (n³ +5n+6) for every integer n ≥ 0.

A: