(i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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

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Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: Let p be a prime. Show that -2 is a quadratic residue modulo p if and only if p = 1 mod 8 or p = 3…

A: Definition(Quadratic Residue Modulo): In number theory an integer qis called a quadratic residue…

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

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Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

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Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

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: Suppose that 2n−1 ≡ 1 (mod n). Is n necessarily prime?

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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…

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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…

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Q: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

A: Want to prove:(a) 10n≡±1 (mod 11) for any n∈N(b) If the integer x has digits xnxn-1.....x1x0 thenx≡0…

Q: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

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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: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p => αp-1…

Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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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: If m = 2* is a power of 2, explain how you could use repeated squaring to compute am (mod n) for all…

A: a) We have m = 2k. First we illustrate the method of repeated squaring to compute am (mod n). Then…

Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

Q: Find the least residue of  -5 -46 modulo 14.

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Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that,                    Expand the right side and show that both sides are…

Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: 1. Show that an integer n is divisible by 2 if and only if its last digit is even. (Hint: For each i…

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Q: a) Show that 2340 ≡ 1 (mod 11) by Fermat’s little theorem and noting that 2340 = (210)34.b) Showthat…

A: According to the given information, Fermat’s little theorem says that:

Q: 7. Let p be a prime number and n= 2P – 1. Prove that 2"-1 = 1 (mod n).

A: Let p be a prime number and n=2p-1. Prove that 2n-1≡1 (mod n)

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

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Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

Q: It is given that 3023 is prime, and 3023 = 2(1511) +1 where 1511 is also prime. (a) How many…

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

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Q: I know that using a corollary that the congruence x2 ≡ 0 (mod p) that the only solutions are + and -…

A: As per the question we have the following congruence relation : n ≡ 2(mod 4) And we have to prove…

Q: Let p be an odd prime. Suppose p5 mod 8. Show p-1 p? – 1 mod 2. 5

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

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Q: . How many different values are there for 4" (mod 7), as n ranges over all positive integers?

A: Note - As per our guidelines we have to answer first question if multiple questions are posted. So…

Q: For every integer n > 1, prove that (n - 1)!mod n = n - 1 if and only if n is prime.

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Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

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Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

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Q: Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

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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: Prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)

A: We have to prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)

Q: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

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Q: Prove that if n is an odd positive integer then 1 + 2 + 3 + · · · + (n − 1) ≡ 0 mod n . What happens…

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Q: 3. Find all primes p < 500 such that there are exactly 18 elements that have order 19 modulo p.

A: Step:-1 Since, we know for a prime p, modulo p forms a cyclic group. And, also according to a known…

Q: 2. Suppose that instead of proving the claim "there is no integer n such that n= 2 mod 6 and n7 mod…

A: consider an integer n=26 such that n mod 6=26 mod 6               =2 mod 6 also, n mod 9=26 mod 9…

Q: What is the smallest positive integer x in the equarion: 1=3x+2(mod 7)?

A: We have to solve the equation 1 = 3x + 2 (mod 7)⇒1 - 2 - 3x = 0 (mod 7)⇒-1 - 3x = 0 (mod 7)⇒3x = - 1…