If p is an odd prime, prove that: 1² · 32 . 52 ... (p – 2) = (– 1)P*1)/2 (mod p) and 22 . 42 . 62 ... (p – 1)? = (-1)(P+ 1)/² (mod p). (p - 1) wwww.

Q: Prove this claim: For every positive integer d and every integer n, the following equation is…

A: Using division algorithm we solve this problem. The steps are given bellow.

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 first three terms of the expansion (4 – 2x)°. 4°(1 – * + + .) 플 + 5x2 + .…) 2 а. b. 25(1 +…

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Q: Let p be an odd prime. Prove that (a) If p = 1 (mod 5) or p = 4 (mod 5), then 5 is quadratic residue…

A: Given p is an odd prime. We have to prove (a). if p≡1(mod 5)  or  p≡4(mod 5), then 5 is a…

Q: 2. Evaluate. (Show intermediate steps for full eredit. You do not need to demonstra application of…

A: Given below the detailed solution

Q: Work the modular arithmetic problem. (9.7)(mod 5). (9.7)(mod 5) = (mod 5)

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Q: b=3(mod19)

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Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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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: Let p be an odd prime. Prove that 2.4.6. (p-5). (p − 3) (p-1) = (-1). 1.3.5 (p-6). (p-4) (p-2) (mod…

A: This question is related to number theory

Q: Using mathematical induction, prove that the Binomial formula holds for complex numbers (²1 Ĺ (2₁…

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Q: 10. (8 points) Solve 3x 2, (mod 7) or show that no solution exists.

A: Solution: The objective is to find the solutions of 3x≡2 (mod 7)

Q: Find a d, he nce ged (2695,154) other wise express 26 95p +15479=d pigtz the in or

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Q: 13. (a) Prove that for each integer a, if a 0 (mod 7), then a #0 (mod 7).

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Q: Work the modular arithmetic problem. (11.10)(mod 7). (11 10) (mod 7) = (mod 7)

A: (a)   (11.10)mod 7  =  110 mod 7                            The remainder when 110 is…

Q: B- State Fermat little theorem, then find √√27 mod 31. Find (52525).

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

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Q: if p. (144) =(-¹) ove tha pand gare distin (p-1)(g-1) 4 aci

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Q: (3) If today is Thursday what day of the week will it be 31 days from today? (4) If it is currently…

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Q: 11- P, X Pz X P, ... X Pn, where M= Pr. Pn are distind primes Prove by inducton, For all Ppositive…

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Q: Determine the cube root of (3 + j4). ound-off the modulus to 3 decimal places, the angle to the…

A: Cube root of (3+4j)

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Q: (a) 8x = 1 (mod 11) by Euclidean algorithm;

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Q: (a) 64x 56 (mod 52) (b) 14x = 35 (mod 56)

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Q: An excercise: Find all the left cosets of in S4

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Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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Q: (a) Prove that 318 = q (mod 19).

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Q: 1) Calculate 1! + 2! + 3!+...+99! + 100! (mod 4). 2) State Wilson's theorem (without proof) Wilson's…

A: (i) To Calculate: (1!+2!+3!+...+100!) (mod 4). (ii) To State: Wilson's theorem.

Q: (a) Let p E N be a prime number, p # 5, and let nE N. Determine the number of solutions to the…

A: First note that this question is only solvable for certain primes and not for all primes. We…

Q: Prove: Let be odd prime. Prove that Aor all i with 1 LikP-2), you have an +... + working so far:…

A: As per the question we just have to show that the prime p divides the following : xi + x2i + ... +…