Statement 1 : 2465 is not a carmichael number Statement 2: 95 = 55 (mod 2) Statement 1 : True Statement 2 : False Statement 1 : True Statement 2 : True Statement 1 : False Statement 2 : True Statement 1 : False Statement 2 : False 71 ansuners

Q: . Show that 1729 is a Carmichael number.

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

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Q: When all the numbers in a given data series appear only once, then the series is said to: a. be…

A: We know that, Mode:     The value occurs with the highest frequency. Uni mode: When there is only…

Q: Statement 1 : 2465 is not a carmichael number Statement 2 : 95 = 55 (mod 2) Statement 1 : True…

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Q: Prove that the fraction 7n+1/ 20n+3  is irreducible for every natural number .

A: Question:-  Prove that the function 7n+12on +3is irreducible far every natural number  →  Let assume…

Q: Congruence conditions (mod 20) for primes p for which -5 is a square (mod p).

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Q: Any two integers are congruent modulo .. when they are both even or both odd. Least common multiple…

A: Given a statement

Q: (a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each…

A: (a) We will prove that  n3≡n(mod3) by the mathematical induction. For n=1 13≡1(mod3) ∴ it is true…

Q: primitive root modulo

A: Given: Value of n=223 in  mod n. To find: Primitive root modulo 223.

Q: prove that tan(ix)= itanh(x)

A: Here we prove thattanix=i tan hx

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

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Q: Using Fermat's Little Theorem find 52003 mod 455. Note that 455 is not a prime number and 455…

A: We are asked to use Fermat's Little theorem to find 52003 mod455. Fermat's Little Theorem states…

Q: Prove: The nth Fibonacci number Fn is even if and only if 3 | n.

A: The nth Fibonacci number Fn is even if and only if 3|n.

Q: Statement 1: 2465 is not a carmichael number Statement 2: 95 = 53 (mod 2)

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Q: Find the smallest positive integer that solves both of the given congruences.   x ≡ 3(mod4​), x ≡ 5…

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Q: ZACB = LECD because of what theorem?

A: Given, Angle ACB= Angle ECD

Q: Let a,n be integers with 2 <n and pod loa,n) = %3D Show that if UIzn) - SanZ;tEZ U(Zn) then ak #1…

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Q: Here are two conjectures: Conjecture 1: For all integers a,b and c, if a|b and a|c, then a|bc.…

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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: 1. Show that 2465 is a Carmichael number.

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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: 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: Let odd prime number P =4 (mod L), be an ond If the integer g iso primitive root with respect to mod…

A: Let r be a primitive root of the prime p with p ≡ 1 (mod 4). We have to show that −r is also a…

Q: Statement 1 : 2465 is not a carmichael number Statement 2 : 95 55 (mod 2)

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Q: (a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each…

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Q: 3 3 tan (80) – tan °(80) Establish the identity. tan (240) = %3D 2 1-3 tan (80) Choose the sequence…

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Q: Use your work from problem 2 and the method used in the example at the end of Lecture 16 to solve…

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Q: Statement 1 : 2465 is not a carmichael number Statement 2 : 95 = 55 (mod 2) Statement 1 : False…

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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: Show that 1729 is a Carmichael number.

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Q: Find a such that 5= 7* modulo 13.

A: Given: 5≡7xmodulo13. The objective is to find the value of x.

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: 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: 1. Show that 1105 is a Carmichael number.

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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: 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…

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Q: Consider congruence mod 5. Let r = {0, 1, 2, 3, 4}. 1) Find r so that 3 =r mod (5). 2) Find r so…

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Q: more general conjecture is that there exists infinitely many primes of the form n^2 + 1; for example…

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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: Suppose n is a large odd number. You calculate 2(n-1)/2 = k (mod),  where k is some integer k is not…

A: Given that 2n-12≡kmodn Take the square on both sides, 2n-12≡kmodn2n-12·2≡k2modn2n-1≡k2modn (a) The…

Q: Determine which of the following congruence relations are true and which are false. (a) 47 = 3 (mod…

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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: Least residue of 63^9 mod 10

A: Rational Numbers: All the numbers which can be written in the form of pq. Here, p and q are integers…

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: 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: (4) (a) State Euler's Theorem or Fermat's Little Theorem. (b) Find the least non-negative residue of…

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.