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 integer n iii. (p- q)2 - 1 = 3 iv. pq is always an even number. (а) i, i (b) i, ii, iii (c) i, ii, iv (d) i, iii, iv (e) All of these

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

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Q: (1) What is gcd(31212021 + 3, 31212021 - 3)? I am aware of 3121 is a prime number, and 2021 is…

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

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: 12 + 22 + ..+n² = n(n+1)(2n+1) 6. is true?

A: Here we have to show that which value of n to proof the mathematical induction in the first step.

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

A:

Q: If a and b are positive integers, which of the following can never be the gcd(56a, 63b)? (а) 56а (b)…

A: given a and b are positive integers,  and to find gcd(56a, 63b) gcd of (56, 63) is 7  so gcd(56a,…

Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

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Q: (Problem 6) Choose ALL correct statements, if more than one is correct. (A) [-5.76] = -5 (В) - 76…

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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: I need your complete solutions. Using mathematical Induction Principle Show n2<n! ; n≥4

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Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

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

A: We need to find the GCD(12345,678)=?

Q: Icm(a,b) = ab if and only if a and b are relatively prime.

A: We know the relation between product of the two numbers their L.C.M and G.C.D Let a and b two…

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: (a) Which primes can divide n² + 1 for some n? a 30000 40 AT & opinip ad seppo 90.4/3 (4) рро t' for…

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

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Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

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Q: (5) Alice and Bob agree to use the prime p = 2663 and the base a = 5 for communications using the…

A: Given  p=2663 , α=5 Let p,α,βAlice be Alice's public key and p,α,βBob be Bob's public key.

Q: expand each expression using the Binomial Theorem

A: Given: 19.(x-2)6

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

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Q: k + 6 k+6 Hint: 126 = -te 3 Assume that n is a positive integer. Compute k+1 k3 + 3k² + 2k* k=1

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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: Q4 (a) Solve for x if 25x = 15(mod 29). (b) Determine the smallest positive integer so that when…

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Q: Arrange the steps in the correct order to prove that 21 divides 4n+1+52n-1 whenever n is a positive…

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Q: Prove by induction that for positive integers n, 13 | (4²n+1 + 3n−1).

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Q: 1) Calculate 1! + 2! + 3!+...+99! + 100! (mod 4). 2) State Wilson's theorem (without proof) Wilson's…

A: (i) To Calculate: (1!+2!+3!+...+100!) (mod 4). (ii) To State: Wilson's theorem.

Q: 2b -126 +4 factors to (36+2). O True O False

A: Consider the statement 8b2-12b+4  factors to 3b+22

Q: 5. Find an integer x between 0 and 776 such that x 13 (mod 21) x = 5 (mod 37)

A: Set N = 21 × 37 = 777. Then , m1  = N/21 = 777/21 = 37 and m2 = N/37 = 777/37 = 21 Now we need to…

Q: Let n = p¹, be a factorization of n into distinct primes p₁,..., Pk. Then, i=1 p(n) = n(1 - 1) (1 -…

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Q: Prove that if p and q are distinct odd primes, then we have (p-1)(q-1) 4 25 52 TID = (-1) quadratic…

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Q: 2(5) – 3(1 – 6) Is even integer?

A: If an integer is divisible by 2, then it is called an even integer. For example, 2,4,6 etc. are even…

Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

Q: For all odd primes p, show that p 2(p-3)!+1.

A:

Statements and Logic

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