Let p be a prime such that p mod 4 = 3. How many elements oforder 4 are in GF(pn)*?

Q: Given 6x + 2 ≡ 101 (mod 7), find an integer z such that 2x ≡ z (mod 7)

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Q: Prove that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

Q: Prove Wilson's theorem that, for any prime p, (p — 1)!= -1 (mod p).

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Q: Show that if p is an odd prime and a and b are integers with a ≡ b (mod p), then

A: According to the given information, let p be an odd prime and a and b are integers with a = b…

Q: 5. If ged(a, n) = 1 then the order of a in (Z/nZ)× is the smallest integer i such that a¹ = 1( mod…

A: Since gcd(a, n) = 1, then             aφ(n) ≡ 1 (mod n)  Now, let order(a) = k.        => by…

Q: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

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Q: Prove that the order of 3 modulo 19 is a divisor of 18.

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Q: Let be a prime such that p = 1 (mod 4). Prove that –1F, is a square in Fp. -

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Q: 3. Prove: for every integer n and for every odd integer a² = 1 (mod 2n+2)

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Q: Let n be an integer greater than 1. If (n-2)! ≡ 1 mod n, then Prove n is a prime number.

A: Use the Wilson theorem and different case to solve this problem as in next step.

Q: (2) Compute the order of a modulo 11 for a = 1,2,..., 10. Find all primitive roots modulo 11.

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Q: Let p be a prime such that p mod 4 = 1. How many elements oforder 4 are in GF(pn)*?

A: It is given that, p be a prime such that p mod 4 = 1. We have to find the number of elements of…

Q: Suppose r is a primitive root mod a prime p such that p = 3 mod 4. Prove ord,(-r) =

A: Given: p≡3mod 4 To find: ordpp-g=p-12

Q: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

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Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: Let n be a positive integer and let ged(b, n) = 1. Show that bx by (mod n) if and only if x=y(mod…

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Q: Is 3 a primitive root of 43? Use your answer to determine all positive integers less than 43 whose…

A: To determine whether 3 is a primitive root of 43, we need to check if it generates all the positive…

Q: Let p be a prime number. Prove that (p -1)! a power p + (3p power2  +1)a is divisible by p, for any…

A: Consider the provided question, (a)

Q: DuppOse that a anа т аге intege Such that a, аre al diпerent IMod- ulo m, with am-1 = 1 mod m. Show…

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Q: Let p be an odd prime number. Prove that [(3)=1] → [p = 1 mod 6).

A: Introduction: An integer-based arithmetic system that takes the remainder into account is called…

Q: Let P be an odd prime. Then for all nЄN, we have () = n²¹ (mod p). Р

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Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

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Q: Find a number a such that 70a = 1 mod 17.

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Q: - Let a, b, n be positive integers. Prove that if a = b (mod n), then (a, n) = (b, n).

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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: Let p be a prime number with p>=5. Prove that p must either be one more or one less than a multiple…

A: We use a fact that any number greater that 5 must be written as one of the form…

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

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Q: Find the integer n that has the following properties: i. gcd(n, 10) = 1 ii. iii. 9 = n38 (mod 10) -9…

A: Concept: The smallest positive integer d that can be a solution or that can be represented as a…

Q: Let p be a prime. Let N be the binomial coefficient (). Show that N = 2 (mod p).

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

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Q: If a is an integer such that: 2 ła, 5ła Show that (a)⁴ⁿ = 1(mod 40) for any positive integer n

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Q: Suppose p is an odd prime. Show that (1² +2 +3P +...+ (p − 1)²) = 0 (mod p).

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Q: 3. Find the last two digits of the number 99°. [Hint: 9° = 9 (mod 10); hence, 99 = = 99+10k. notice…

A: In question number 3: we have to find the last two digits of the number (((9)^9)^9) That means to…

Q: Show that if n is prime and b is a positive integer with n ̸∣ b, then n passes Miller’s test to the…

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Q: e) Let p and q >7 be primes such that 2 +2P +1 =0 mod q. Prove that q = 1+6kp for ome natural number…

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