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Q: Let x and y be positive integers such that the prime factorization of x has exactly five 3's and the…

A: x = 35×k          k is an integery = 33×l           l is an integer

Q: Let x be an integer coprime to 1450 that is not congruent to 1 modulo 1450. Suppose we know that x1…

A: The given two MCQs are exactly same. Hence we will just solve the first one. And we will use the…

Q: One number-theoretic result mentioned in the course was Wilson’s Theorem: If p is a prime then (p −…

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Q: #3 please

A: The Fibonacci numbers for n = 1, 2, …, 18 are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…

Q: Let p be a prime larger than 3. 1. Show that the remainder, when p is divided by 6, is 1 or 5. Note:…

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Q: If p and q are twin primes, then which of the following is true. i. p + q is divisible by 3 ii. pq =…

A: We have to solve given problem:

Q: If p and q are twin primes, then which of the following is true. i. p + q is divisible by 3 ii. pq =…

A: twin prime numbers are the prime numbers which are separated by difference of 1 for example 11 and…

Q: (a) Give an inductive proof that the Fibonacci numbers Fm and Fm+1 are reatively prime for all m 20.…

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Q: Suppose that p is a prime and p ≠ 2. Let a be a nonsquare inGF(p)—that is, a does not have the form…

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Q: 111111 is prime. Prove that, in any radix b, The number 11111 (11...1), can be prime only if the…

A: As per the question we have to prove that the number 111...1 containing n number of 1's will be…

Q: gcd (k, Pal)=- x* =b (mod P). Theorm: If p aned q are dis tinbed primes non of which divides b, and…

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Q: (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X? + 3 has roots in Z/61Z by (a): they…

A: Here we need to solving ....(i) knowing that .....(ii)

Q: Is the following statement true or false? For every integer n, if n is prime then (-1)^= -1. Which…

A: The given statement is: For every integer n, if n is prime then -1n=-1. We know that a non negative…

Q: (g) There are exactly six prime numbers p with the property that x81 = x (mod p) for every integer…

A: Given that, x81≡xmod p Find the six prime which satisfies the given congruence. Fermat's Theorem: If…

Q: Theorem 1. For every prime p, /p is irrational. Proof. Write your proof here... Theorem 2. Let a, b,…

A: NOTE:- By bartleby rules, only first question is answered To prove that for every prime p, p is…

Q: Theorem 1. Every positive integer n > 2 can be written as a product of primes. In other words, for…

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Q: Use Euler's Theorem to find 5134 (mod 23) 5134 = 25(mod 23) None of the choices This option This…

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Q: Given n is a positive integer, prove that n is even if and only if 3n + 5 is even.  You must use…

A: Topic:- algebra

Q: 2. Find the Eurler's function of 45, i.e., (45) =? 3. For n EN and p is prime, prove that (nlp - 1…

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: .Find the flaw in the following proof. JECTURE: If n is a prime number, then 2"- 1 is also prime.…

A: Please concentrate when you  see  this  answer.

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: Only one prime of the form1+4n exists. Determine this prime number and prove it's the only one of…

A: A number that is greater than 1 is said to be prime if it has only two factors 1 and the number…

Q: 11 Jlg Let a and b be integers, not both zero. Then a and bare relatively primeif and only if 1 = ax…

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Q: Euler's Theorem states that if n is a positive integer and a is relatively prime to n, then ar(n) =…

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Q: the congruence + 92 (2), using the method (Bertrand's Postulate). If n > 0 there erists a prime p…

A: Given:- x4-2x3+9x2+4≡0 (mod 72) , using the method (Bertrand's Postulate).  if n>0 there exists a…

Q: Find a proper factor p of 267 – 1 using the point P = (6,7) on the curve y? = x +6x – 203. State the…

A: Use Fermat's little theorem to solve the following question. Assume different values to deduce the…

Q: If the gcd(p, q) = 1, then p and q are said to be relatively composites relatively primes primes…

A: Composite Number -  A number that has more than 2 factors is called a composite number. Prime Number…

Q: Show that for any odd prime p, the Gauss, G(p), can be rewritten as p-1 G(p) = 5². x=0

A: To Show- Show that for any odd prime p, the Gauss, Gp, can be rewritten as Gp = ∑x=0p-1ζpx2.

Q: Find s e Z with 0ss< 13, so that s = 127+300(mod 13). Your work should show a method that does not…

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Q: 28. Show that if p and q are distinct primes, then p-1+qP-=1 (mod pq).

A: To prove: If pand q are distinct primes then, pq-1+qp-1≡1(mod pq).

Q: 7 .Assune p is an odd prime such that p ≡ 1mod4 (that is p = 4k+1) Prove: if a is a primitive root…

A: Given: p≡1mod4       ∵p=4k+1

Q: (8) Using that 151 is a prime (so that, in particular, Fermat's Little The- orem can be used; hint,…

A: That's easy. Have a great day!!!

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

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Q: 1. If a is an odd integer, then a2 +3a+5 is odd. (use direct proof) 2. Suppose x and y are even. If…

A: Given:1. If a is an odd integer , then a2+3a+5 is odd2. Suppose x and y are even.If x i even, then…

Q: Find a and b that make the following statement true, and prove your answer. Let p be an odd prime.…

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Q: 3. Find integers s and t so that 1 7s+ 11t. Show that they are not unique. %3D

A: Apply the division algorithm, we get 11 = 7*1 + 4 7 = 4*1 + 3 4 = 3*1 + 1 Hence, 1 = 4 - 3*1   = 4 -…

Q: (2) Let p be an odd prime and suppose that m e Z is not divisible by p. (1) Show that E() \p- = 0.…

A: (1) Given that p is an odd prime. The aim is to show that ∑a=1p-1ap=0. Now take, ∑a=1p-1ap,…

Q: Theorem 4.10. Let p be an odd prime. Then among 1, 2, ,p-1, exactly (p-1)/2 many are quadratic…

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Q: Assune p is an odd prime such that p = 1mod4 (that is p 4k + 1) Prove: if a is a primitive root for…

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Q: 4. Given that 5 is a primitive root of 17, find the smallest positive integer x such that 6* = 10…

A: As per our company guidelines I am supposed to do only the first question kindly repost the…

Q: Problem 3. Find all the positive primesp such that 9p +1 is a perfect cube. Namely, such that there…

A: A prime number is such a number which is divisible by 1 and the very number. Prime numbers play a…

Q: Which of the following is true? There are infinitely many primes of the form n2- 9. If lcm(a, b) = b…

A: there are given options. there are infinitely many prime of the form n2-9 it is not true as…

Q: 2. Show that if a and b are positive integers and alb, then a < b. 3. Show that if a and b are…

A: Basic facts about division will be used in solving this problem.

Q: Show that if m is an integer then 3 divides m3−m.

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

Q: (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X2 +3 has roots in Z/61Z by (a): they…

A: Given that, 61 ≡ 1 mod (12) The polynomial x2 + 3 has roots in Z/61Z. The roots are 27 + 61Z and -27…

Q: Fermat's Little Theorem tells us that if p is prime and a is an integer not divisible by p, then…

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Q: 19. (Congruences) Let n and a be fixed positive integers and let d = gcd(a,n). Show that the…

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Q: Fermat's Little Theorem says that for all natural numbers p and integers x, if p is prime then x = x…

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