Fermat's "Little" Theorem states that whenver n is prime and a is an integer, d"-1 = 1 mod n a) If a = 133 and n = 157, then efficiently compute 133156 = 1 mod 157 b) If a = 25 and n = 97, then efficiently compute 25 · 2595 = 1 mod 97 Use the Extended Euclidean Algorithm to compute 25- = -31 mod 97. Then 259 = 8 mod 97. c) If a = 152 and n = 463, then efficiently compute 152464 mod 463 II

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: Suppose that p, q are distinct prime numbers, prove that p9-1 + gp-1 = 1 mod pq

A: We use Fermat's Little Theorem, to prove our claim.

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

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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: 2221 (mod 13) is equal to O 10 (mod 13) O 1 (mod 13) 7 (mod 13) 6 (mod 13) O None of these

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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: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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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: 6. Trace Algorithm 4 when it is given m = 7, n = 10, and b = 2 as input. That is, show all the steps…

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Q: Find the smallest non negative value of D such that ED is congruent to 1 (mod b) when b=616…

A: As given in the hint the idea is to use the extended Euclidean algorthim to solve the congruence.…

Q: 2) Find the smallest positive integer z, such that z = 13 mod 4 z = 20 mod 9 z = 11 mod 25 .

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Q: Compute 21200 mod 27.

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

A: Given, Fermat's Theorem :- If n is a prime number and a denotes an integer such that a,n=1, then…

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…