38. It is known that 14 is a primitive root for the prime p = 30000001. Let 14⁹000000 (mod p). (The exponent is 3(p − 1)/10.) b = 1. Explain why 6¹0 = 1 (mod p). 10 2. Explain why b ¥ 1 (mod p).

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: 1. Homework. Use the Arithmetic Modular operations to find 55552222 mod 7. Hint: 5555 = 4 (mod 7).

A: Details are attached in step 2.

Q: True or false: If p is an odd prime, then (p-1)! + 2^(p-1) congruent 0(mod p). Justify your answer.

A: p is an odd prime.

Q: Use Fermat's Little Theorem to find each value. Remember, x-1) =1 (mod p). Write your answer in…

A:

Q: Find the value of 592 (mod 11) using Fermat's Little Theorem. Remember, x (P-1) = 1 (mod p) a 592 =…

A:

Q: "divisionless mod" where x mod y is computed in steps. One step is to use xhi, xlo, a and b to…

A: xhi =14 , xlo  =4 , a=5 and b=3 To compute x mod y.

Q: for 3^800 how can I find the last digits of it by using mods (such as last digit and 2nd last…

A:

Q: Suppose that a and b are integers such that a = 45 (mod 71) and b = 53 (mod 71). Find an integer c…

A:

Q: Find the residue for the following case. 86 (mod 7) The residue of 86 (mod 7) is (Round to the…

A:

Q: 3. Find the following: A. 3^0 mod 5 1 B. 3^1 mod 5 3 C. 3^2 mod 5 1 D. 3^3 mod 5 4

A:

Q: 2. Find a primitive root modulo 41.

A: We have to find a primitive root of 41.

Q: 2. Determine x >0 for each case: 5903 = x (mod 60), 17342 =r (mod 5).

A:

Q: What is the largest value that m mod n can be?

A: Largest value of m mod n is n-1.

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: Show that every nonzero residue (mod 17) can be written as a power of 3

A:

Q: 3 Р 1 ? What if 3 is replaced by = 2. For which primes p is -3, 5, -5, 6, or -6 ? Any patterns? Is…

A: The symbol  is the Legendre symbol, used in number theory to denote whether a number is a quadratic…

Q: # 14 please ASAP

A: We need to evaluate 77 (mod 4).By the Fermat’s Little Theorem,

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)

A:

Q: (d) (e) (f) Assume that a 15 = 1 Find the value of ord41a. mod 41 and a #1 mod 41. How many…

A: As per the question we have to find : ord41(a) , where a15 ≡ 1 mod 41 but a ≠ 1 mod 41 total number…

Q: how do i use Euler's thm to find the value? 3242 mod 385

A:

Q: Let a, n be positive integers. Knowing that 15a - 13n = 1, we deduce that the multiplicative inverse…

A:

Q: Identify if the given statement is true or false: I. If a is an integer, the (a^2 is congruent to 0…

A: Given some statements. Check whether these statements are true or false.

Q: True or False: 3^208= 3 mod 53.

A:

Q: If a = 17 (mod 3), then what is the remainder of a when divided by 3? Enter the numerical value.

A:

Q: A) Find the multiplicative inverse of 4 modulo 13. B) Use your answer from (a) to solve 4x = 5 (mod…

A: a) To find the multiplicative inverse of 4 modulo 13. b) To solve the linear congruence 4x≡5mod 13.

Q: (a) Find all solutions r to the congruence 341x = 16 mod 912

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

Q: For which p= 3 mod 4 is i a square in F»[i]? For which p = 3 mod 4 is i the square of an element in…

A:

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

A:

Q: 1. Solve the congruence x² + 1 = 0 mod 13³.

A: To solve the congruence  , we can use the fact that  is a prime number.The congruence  implies that…

Q: If x00 = 1(mod 109) then possible value of x is a) 108 (b) 218 (c) 109 (d) None of these

A:

Q: Hanb(mod n) and b c(mod n) then which of the following is trae. (a) anc (mod n) () 2=c(mod n) (c)…

A:

Q: whether 2 is a primitive root modulo 19. Please state the reason.

A:

Q: (d) 357 862 193 = (mod 943). (e) 5327 6135 7139 2187 5219 . 1873 = (mod 8157). (Hint. After each…

A: ...

Q: Compute (1001 mod d) for d = 2, 3,..., 10.

A:

Q: Suppose that a and b are integers, c b = 3 (mod 19). Find the integer c that a) с%3D 13а (mod 19).…

A:

Q: 20.What is the value of n such that (24  is congruent to 58 mod n) ? a.13 b.17 c.-34 d.-13

A: # as per the guidelines we are entitled to solve one question at a time, please resubmit the other…

Q: Show that every nonzero residue (mod 17) can be written as a power of 3.

A: We show that every non zero residue(mod 17) can be written as power of 3.

Q: You are trying to factor n = 2288233. Suppose you discover that 8805252 = 2 ,  20572022 = 3, and…

A:

Q: 1. i) Explain why 3 is a primitive root modulo 14 ii) What is log3(11) mod 14?

A:

Q: 5. Homework. Use the Arithmetic Modular operations to find 1929570 (mod 27) +231500002 (mod 24)

A: To solve this problem, we need to use modular arithmetic properties.   Let's start with the first…