Prove or disprove. There is a prime of the form p = 6k – 1 such that the congruence 13x? + 7x +1 = 0 mod p is solvable.

Q: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

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Q: a p−1 ≡ 1( mod a) for any integer a and prime p. Select one: True False

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Q: Let p ≥ 2 be a prime. Suppose p does not divide a. Then ap−1 ≡ 1 (mod p).

A: The given problem is to show the given following modular congreunce that let p is a prime and p…

Q: How do you fiNd all positive integers m for which the following congruence is true. 100 ≡ 50(mod m)

A: We know if a≡b( mod m) then a-b = mk where k is integer. Given,      100 ≡ 50 ( mod m ) We have to…

Q: Is it true that (3 x EN) (x²-3x +20 is a prime)? O True False

A: Is it true that ( there exists x in N)(x^2-3x+20  is a prime ) ?   True False

Q: Prove that if gcd(a, m) =1 and gcd(a – 1, m) =1, then 1 + a + a^2 + ....+ a^ phi(m)-1 congruent to 0…

A: Given gcda,m=1 and gcda−1,m=1 We have to show that 1+a+a2+......+aϕm−1≡0modm Since, gcda,m=1⇒m/a And…

Q: and cto (mod P) 37. Prove that if p is a prime and C#o (mod p), solutien then cx=b (mad p) has. a…

A: We need to  Prove that  p is a primeand  c ≢  0 (modp);then cx≡ b (mod p) has a unique solution

Q: If p is a prime with p = 5 mod 6, then (²) O -1 O None of these

A: If p≥3 is an odd prime, then    (-3/p) = 1 if p ≡ 1 (mod 6)    (-3/p) = -1 if p ≡ 5 (mod 6)

Q: If p > 2 is a prime of the form 2m +1, prove that m is a power of 2.

A: Let p be a prime and p > 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: a) Let p be an odd prime. Show that a = -1 mod p has a solution if and only if p = 1 mod 8.

A: Consider the provided question,Hello. Since your question has multiple parts, we will solve first…

Q: 4. Let p be an odd prime, and a, b primitive roots mod p. Show that ab is not a primitive root mod…

A: It is given that, p be an odd prime, and a and b primitive roots mod p.

Q: Prove Euler’s criterion, which states that if p is an odd prime and a is a positive integer not…

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Q: {-74 + 1, ..., 74 – 1}. (b) Hence, find a solution to the congruence x' = 99 mod 74.

A: Given:- An expansion above Polynomial will have co-efficient lies between {-74+1,..........,74+1}

Q: a) How can an inverse of a modulo m be used to solve the congruence ax ≡ b (mod m) when gcd(a,m) =…

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Q: Prove that if p is a prime and a is any integer so that p does not divide a, then the additive order…

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Q: Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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Q: *discrete math* Prove that for all integers a, b, if a and b are congruent to 1, mod 3, then their…

A: An integer n is congruent to 1, mod 3, if and only if there is an integer k such that n = 3k + 1.

Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: 3. (а) Using the Legendre symbol, determine your own prime number Pi and non-zero coefficients…

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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: 5x = 7 (mod 32)

A: To solve the following problem

Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p > 5. p2−1≡0mod24

Q: brvib - If m is composite, what is the value (m – 1)! mod m in the standard residue system? Conclude…

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Q: If p is an odd prime, prove that: 1² · 32 . 52 ... (p – 2) = (– 1)P*1)/2 (mod p) and 22 . 42 . 62…

A: Given that p is odd prime. (i) To prove 12.32.52....(p-2)2≡-1p+12modp Since,…

Q: If a and b are positive integers and p is prime with p ł a, then the solutions of the linear…

A: We will check whether the given statement is true or false.

Q: Use Hensel's Lemma to find a solution to the following congruence: a +3 x + 3 = 0 mod 54.

A: Let f(x) be a polynomial , p is a prime number . If x=a is a solution of congruence equation…

Q: (a) Find an r E Z that makes the following congruence true: 5x = 1 (mod 24). (b) Prove that there is…

A: Solution a: Given  5x≡1 mod 24 For x=1 5≡5 (mod 24) for x=2 10≡10 (mod 24) for x=3 15≡15 (mod 24)…

Q: Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).

A: we will prove that gcd (a, n) = 1, which  means that a and n are  prime. Let m= gcd (a, n) = gcd (a…

Q: Let n = 2p for some prime number p > 2. Show that if a is a positive integer such that gcd(n , a)…

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Q: 9. Let p be an odd prime, and let ( - 1)(a – 2).. (a - (p - 1)) = "--g12-2 -3 -...+ ap-32 - p-2a +…

A: Let a be any solution of x^2 + 1 ≡ 0 (mod p), so that a^2 ≡ −1 (mod p). Becausep doesn't divide a.…

Q: (a) Let p E N be a prime number, p # 5, and let nE N. Determine the number of solutions to the…

A: First note that this question is only solvable for certain primes and not for all primes. We…

Q: If a and b are primitive roots modulo a prime prime p, then ab is a primitive root modulo p. O True…

A: Answer  ; False Explanation

Q: Let p be prime. (a) If a 1 (mod p), show that 1+ a + ...+ aP-1 = 1 (mod p). (b) If a -1 (mod p),…

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Q: The preceding part of the problem proves the reverse implication unless n = p2 where p is a prime.…

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Q: 9. is a prime ideal in C[x] a) True b) False O a) True O b) False

A:

Q: (a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b)…

A: (a) Given that p is an odd prime.  The quadratic formula mod p is given by: ax2+bx+c≡0mod p Here, p…

Q: Let a and b be integers and n a positive integer. Assume also that a and n have a common divisor d…

A: Use the following concepts, to prove the required result. If a divides b then b is a multiple of a.…

Q: 2. Find all solutions, or show there are none, to the congruence X78 = 5 (mod 19)

A: By Fermat little theorem x18≡1mod19. Therefore, x4×18+6≡x6mod19 or x78≡x6mod19.

Q: If p and q are distinct primes and a is an integer such that aº = a(modq) and aª = a(modp), then

A: This is a problem of theory of congruence, number theory.

Q: The number of incongruent primitive roots modulo a prime p is: a) Ø(p) b) Фф — 1) с) р—1 O a O b

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Q: Use induction on n to prove that 42n – 15n – 1 is divisible by 225 for all non-negative integers.

A: Prove 42n-15n-1 is divisible by 225 for all non-negative integers using mathematical induction.

Q: Prove both subparts  (i ) Prove that if p is a prime, then (p − 1)! ≡ −1 (mod p). (ii ) Show that…

A: If P is prime then to show that (p − 1)! ≡ −1 (mod p). And 2nd part (ii) we prove using the first…

Q: Prove that for every odd prime 1000 < p < 2000 the number of –2, –1,2 which are squares mod p is not…

A: To prove that for every odd prime numbers strictly between 1000 and 2000 the numbers -2,-1, 2 which…

Q: 2. Let p and q be distinct primes. Let a be an integer such that aº = a aª = a mod p. Prove that aPª…

A: ap = a (mod q) and aq = a (mod p)

Q: Suppose a and b are relatively prime positive integers. Is it true that (a+2b)4(ab) = 1 (mod ab) ?…

A: We try to solve the question using Euler's Theorem. The detailed solution typeset in LaTeX is…

Q: 6) If p is an odd prime, then 1P + 2P + 3P + .. + (p – 1)P = ??(mod p) a) 0 b) 1 c) - 1 d) None of…

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Q: 3. Euler made the following conjecture: Let a be any natural mumber and p, q primes such that p= ±q…

A: Let a be any natural number....and p,q be odd primes...

Q: e ab then th

A: Given: ak^2+bz^2 =c mod p To prove the given.

Q: There is an integer n such that 7n2 – 5n + 11 is a prime number. -

A: Let fn=7n2-5n+11for n=1f1=7×1-5×1+11      =13 , which is prime number for n=2f2=722-5×2+11…

Statements and Logic

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