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

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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: 1/sqrt(1) + 1/sqrt(2) + . . . + 1/sqrt(n) = sqrt (n) for n=1,2,3

A: We have to prove  11+12+13+...+1n≥n For n=1 11=1 For n=2   Suppose for n=k the statement is true…

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: 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: 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: (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: (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…

A: Fermat's little theorem is one of the most important and significant methods in number theory.…

Q: (18)280 mod 125 =

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

A: We have to solve  by Euclidean algorithm.Note: According to the guidelines we are allowed to solve…

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

A: We have been given S4, a symmetric group. We have to find all the left cosets of <1 2 3 4>in…

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: 10 Solve 3Y=2 (mod 7) 8 Salutions exist or show that

A: We have to solve given problem:

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 + ... +…