Let n be a positive integer, and a an integer Prove that if the order of a modulon is n-1 then n is a prime

Q: Let p be a prime and k a positive integer such that ak mod p = a mod p for all integers a. Prove…

A: Consider the provided that, ak mod p=a mod p,                ....1       where p is prime and k is a…

Q: Prove that : If p is a prime number and p does not divide a then a- 1(modp).

A:

Q: Let m be a positive integer. Show that a mod m = b mod m if a ≡ b (mod m)

A:

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

A:

Q: |Let m be a positive integer. Let a and b be integers such that a = b (modm). Use mathematical…

A: Solution

Q: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: 1. Let n = pq, where p and q are distinct odd primes. Define X(n)= (p-1)(q-1) gcd(p-1,q-1)* Suppose…

A:

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: a be an odd integer. Pro

A:

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: (a) Let n be any integer. Prove that n2 is congruent to 0, 1 or 4 modulo 8.

A: Note: According to bartleby guidelines we have to answer only first question please upload the…

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

A:

Q: Let n be a positive integer, and a an integer Prove that if the order of a modulo n is n-1 then n is…

A: Problem 2.  Solving it using the concept of phi function

Q: Prove that o – 1 is divisible by 4.

A: Even numbers are numbers divisible by 2. Odd numbers are numbers that are not divisible by 2. The…

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

A:

Q: 5. Let p be an odd prime, and suppose that ged(a, p) = 1. Show that the congruence z = a (mod p*)…

A: p is an odd prime and gcd(a,p)=1We have to show the congruence x2=a mod pk has exactly 1+ap…

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: Number theory

A:

Q: Let p be an odd prime, and let g be a primitive root modulo p. Prove that -g is a primitive root…

A: Given : p= odd prime g= primitive root modulo p

Q: Show that if a,b, and n are integers such that n≥2 and a≡b (mod n), then gcd(a, n) = gcd(b, n).

A:

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: 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: Let p = 2^k + 1 be a prime for some k ≥ 1. Let a be any integer such that gcd(a, p) = 1 and a is NOT…

A: Consider the given information:

Q: How to solve this?

A: According to the given information, let p be a prime number such that p>2. It is required to…

Q: 1. Let p be an odd prime. Let a be an integer such that x2 = a (mod p) has that a (mod p has exactly…

A: 1. Let p be an odd prime. Let a be an integer such that x2 = a (mod p) has that a (mod p has exactly…

Q: (a) Show that o(668) = 20 (ø (668)). (b) Prove that if p is a prime greater than 2 then 2(p – 3)! =…

A: We will apply Euler Totient function for part a.

Q: Let n = pq, where p and q are distinct primes such that p = q = 3 (mod 4). Suppose a € Z+ with a <n.…

A: given data, n=pq , where p and q are distinct primes such that p≡q≡3(mod 4). also aϵZ+ with a&lt;n.…

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

A:

Q: * Exercise 3-2.(a) Prove that if m = 2p, where p is an odd prime, then (m – 1)! 0 mod m².

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

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: 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),…

A:

Q: If a has order n-1 mod n, then n is prime. O True O False

A:

Q: Show that the Bernoulli number B₁ satisfies B₁ = (-1)" ( 2 + 2d²+1) Σ (mod 1) for any positive…

A:

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: 30. Prove that any positive integer is congruent to its units digit modulo 10. 31. If a = b (mod n),…

A:

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

A:

Q: Let n ∈ N and p be a prime. Prove that if n is not divisible by p then n and p arerelatively prime

A: From Prime factorization theorem we know that any natural number can be written as product of…

Q: (a) Assume that n1, n2 and n3 are integers with gcd(n1, n2) = gcd(n2, n3) = gcd(n1, n3) = 1. Let…

A:

Q: Let p be an odd prime and a ∈ (Z/pZ)x be such that (a/p) = 1. Show that a is a square modulo every…

A: Given: Let p denotes an odd prime and a ∈ (Z/pZ)x be such that (a/p) = 1.

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

A:

Q: Every natural number n has a base-60 expansion; that is, there exists k > 0, and for each i = 0, 1,…

A:

Q: 4. Definition: Let M be a modulus greater than or equal to 2, and A an integer. If A" = 1(mod M) for…

A: Solution a: order of 4 Here 41≡4 (mod 13)42≡3 (mod 13)43≡12 (mod 13)44≡9 (mod 13)45≡10 (mod 13)46≡1…

Q: Let n > 2 be a positive integer and r a primitive root of n. Suppose that a E Z is such that ar? = 1…

A: Given: r is a primitive root of n. To find: ϕn2

Q: Prove by induction on n that, for all positive integers n: Σο (i(i!)) = (n + 1)! – 1

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

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: 6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not…

A: The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…

Q: Use mathematical induction to prove the following statement. If a, c, and n are any integers with n>…

A:

Q: Let n > 2 be a positive integer and r a primitive root of n. Suppose that a E Z is such that ar? 1…

A: