gcd (k, Pal)=- x* =b (mod P). Theorm: If p aned q are dis tinbed primes non of which divides b, and iR gcd (k,cpol), cq-1)]=L, t then x =b (mod pq) has a unique Solution modulo

Q: Prove that there are infinitely many primes congruent to 3 mod 8. previous part.

A: We have to show that there are infinitely many primes congruent to 3 mod 8. we will proof this by…

Q: Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute…

A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴  b − a = −17k = 17(−k); henceb ≡ a (mod…

Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

A:

Q: TRUE or FALSE: 1. If a is congruent to b (mod m), then a^k is congruent to bk (mod m) for every…

A:

Q: Fermat's "Little" Theorem states that whenver n is prime and a is an integer, a"-l = 1 = 1 mod n a)…

A: Given a =463 and  n=953 We need to find x such that                   463952 ≡ x (mod 953).

Q: Show that if p is a prime number then every equation of the form ax ≡ b mod p with a 6≡ 0 mod p has…

A: Given p is a prime number and 6≡0 mod p. Therefore either p=2 or p=3. Let d=gcda,p. We know that if…

Q: Give an odd prime p congruent to ±1 modulo 12. Show that (3) = 1.

A: Please see the attachment

Q: If p is a prime with p=3 mod 4 and a is quadratic residue mod p, then -a is a quadratic residue mod…

A:

Q: Find all integer solutions satisfying all the equations given. 5x =1 mod 39 3x =1 mod 25 4x =1 mod…

A: Concept :.   *   ax =6 mod n                           g (d (a,n) = d (let)…

Q: Prove, that if 2m + 1 is a prime number, then m must be a power of 2 (m = 2n, for some n ∈ N). Prime…

A:

Q: Alice and Bob want to use Diffie-Hellman Key Establishment to share a key and they have agreed to…

A: The suitability of these numbers in the order is shown as given below

Q: 8. (a) Prove that if p and q are odd primes and q | a² - 1, then either q |a - 1 or else q=2kp + 1…

A: a) Given that p and q are odd primes and qap-1. So, ap≡1mod q If p=1 then  a≡1mod q So, qa-1 If p≠1…

Q: Let be a prime number. Stating any results that you use, deduce that xº-1 = (x – 1)(x – 2) · . · (x…

A:

Q: Y OU (a) Find the least residue of 639 modulo 10. (b) Use Fermat's little theorem to find the least…

A:

Q: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: (9) Find all prime numbers p for which both equations X²-2 0 mod p and have solutions. X²-3 0 mod p

A: Concept: The quadratic congruence: x2≡a(mod p) , where gcd(a,p)=1 Has a solution if (ap)=1 That is a…

Q: (5) Show that if n ɛ Z+ is odd which is divisible by k distinct primes, then the congruence x2 = 1…

A: Chinese Remainder Theorem: If a1,a2,...,ak are integers, and m1,m2,...,mk are pairwise relatively…

Q: Suppose that residue(modp) 8) Let odd prime P. atzis Show that has be an not Diophantine…

A: Given that a is not a quadratic residue mod p. That is there does not exists an integer z such that,…

Q: 6. Show that ifn = pq, a product of distinct primes, then a (n)+1 (mod n) for all a. = a

A:

Q: 1. Suppose m, nEz* are relatively prime. Prove that for all a, bEZ, a =b (mod mn) if a =b (mod m)…

A: Note that: a is congruent to b modulo m if and only if m divides (a-b).

Q: Suppose x satisfies the three simultaneous congruences X = 4 (mod 5) x = 7 (mod 8) X 10 (mod 11)…

A:

Q: 9. Show that if p is an odd prime and p = 3 mod 4 then p is not the sum of the squares of two…

A: In the given question we have to prove that if p is an odd prime and…

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: Least residue of 45^37 mod 19 when using Fermats Little Theorem

A:

Q: 6. Let Fn 22" + 1, n > 1, be a prime Fermat number. Show that 3 is a primitive root modulo Fn.

A: Given: Fn=22n+1,n>1

Q: 6), then the congruence has a unique solution modulo p. 11. Show that the congruence x³ = 3 (mod 19)…

A: 11. To prove that the congruence x3≡3 mod 19 has no solutions, whereas x2≡11(mod 19) has three…

Q: 2. Find the largest negative x € Z that simultancously satisfies 4x = 9 (mod 5) 2x = 0 (mod 6) x =0…

A: Given : 4x = 9 (mod 5) 2x = 0 (mod 6) x = 0 (mod 4)

Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

Q: If possible, let berational. v2 a Then, there exist positive co -primes a andb such that - -

A:

Q: You and your friend Lara are considering the linear congruence ax = b (mod m), where m = 6 x 11 x…

A:

Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

Q: Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no…

A:

Q: Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)

A: I have proved it by using defination of congruent.

Q: Prove Exercise 4.6.12 in the course notes. Then, assuming that gcd(a, n)|b, determine how many…

A: We have to prove that if gcd(a,n) | b, then ax≡b (modn) has no solution. We will prove this by the…

Q: A.)Many printed airline tickets contain a 10-digit document number followed by a check digit. The…

A:

Q: 1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not…

A: As per BARTLEBY GUIDELINES , I answered one question (i.e. Q-1) and repost other questions…

Q: Let p be prime. Show that C) mod p = C for 0 <k< p What does this imply about the binomial…

A: See the following below.

Q: Prove that there are infinitely many prime numbers that are square modulo 11 (trying to prove…

A: To prove: There are infinitely many prime numbers that are square modulo 11.

Q: Find all solutions to the congruence 10x≡5 mod 15 (b) Prove that 7 divides 32n+1+ 2n+2for any…

A: In  (a) we use the extended Euclidean algorithm to solve the problem. In (b) we use the mathematical…

Q: 2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and…

A: Fermat's Little Theorem:- It states that if p is any prime and a is any integer then, ap≡amodp…

Q: Determine a general congruence condition satisfied by all primes p with p = 3(mod4) such that the…

A: As per the question we are given a number theory problem. Where we have to find a congruence…

Q: 3. a) Compute Ek. b) Prove that there are infinitely many prime numbers. c) Assuming modulus 127,…

A: Given: The following conditions. To find: a) Compute ∑k=0nk3 b) Prove that there are infinitely many…

Q: Suppose ab≡0 mod n and gcd(a, n) = 1. Apply Euclid’s Lemma to prove b≡0 mod n.

A: Given: ab≡0mod n and gcd(a,n)=1

Q: 3) Support the p is a that Show least one Prove then iss prime 27. ท (1^2) = (2+1) = 1 for at number…

A: Disclaimer:  “Since you have asked multiple question, we will solve the first question for you. If…

Q: 1- Find a solution of 621m + 483n = k, where k is the gcd of 621 and 483.

A: 1. Given that, the equation is 621m+483n=k and k is the gcd of 621 and 483. Use the Euclidean…

Q: (a) gcd(q, n) = 1 if and only if kq # 0 mod (n) for all 1 < k< n. %3D

A:

Q: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False

A: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False

Statements and Logic

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