Fermat's "Little" Theorem states that whenver n is prime and a is an integer, d"-1 = 1 mod n a) If a = 29 and n = 659, then efficiently compute 29658 %3D mod 659 b) If a = 102 and n = 102 · 102177 = 179, then efficiently compute mod 179 Use the Extended Euclidean Algorithm to compute 102- = mod 179. Then 102177 = mod 179. c) If a = 406 and n = 509, then efficiently compute 406510 mod 509

Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: Using Euclid's algorithm, find the multiplicative inverse of 23 modulo 75 and then solve the linear…

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Q: Please show me the necessary steps to prove this. Thanks!Solve the congruence 156x ≡ 402 (mod 570),…

A: In second page " . " means multiplication (in case if you get confused)

Q: Prove that 2 is not a primitive root modulo 31.

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Q: Use Fermat's Little Theorem to compute 51990 mod 7.

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Q: Find the order of all element of Z8 under addition mod . Please brieflt explain and show all the…

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Q: Find the inverse ā of 5 modulo 187 using (extended) Euclidean Algorithm with 1 <a< 186.

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Q: Compute 17^7121 (mod 11).

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Q: Use square and reduce algorithm to compute 12377(mod 322). Show that the answer is 93.

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Q: a) For an odd prime p, prove that the quadratic residues of p are congruent mod p to the integers…

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Q: 3) Use the Euclidean algorithm to find (662,488) and express it as a linear combination of these…

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Q: Compute 25200 mod 27.

A: As 25 is coprime to 27, by Euler's theorem, we see        25φ(27) ≡ 1 (mod 27)  Now,        φ(27) =…

Q: Find all integers between 0 and 20 that are a perfect squares of mod 21

A: 4, 9, 16 only three numbers between 0 and 20 whose perfect squares are integers under mod 21 1) √4…

Q: Compute 177121 mod 11.

A: To find 177121 = mod 11

Q: Fermat's "Little" Theorem states that whenver n is prime and a is an integer, d"-1 = 1 mod n a) If a…

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Q: Compute 21200 mod 27.

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Q: Use the main theor 't take powers)

A: We have to show that 12 must be the primitive root of 25 by using the main theorem on the primitive…

Q: Use the Euclidean Algorithm to determine if 7^-1(mod 20) exists.

A: To determine if 7-1mod 20 exists, using Euclidean Algorithm.

Q: Show that (-3) = 1 if q = ±1 mod 12 -1 if q = ±5 mod 12

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Q: answer only for d)

A: gcd(1,769,2,378)=1,769*x+2,378*y2,378=1*1,769+6091,769=2*609+gcd(1,769,2,378)=1,769*x+2,378*y2,378=1…

Q: show that 10 is a perfect square mod 45 and give at least two square roots (between 0 and 44)

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