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: 7) Show that there are no primitive roots modulo 21 using EGLF and CRT.

A: Given value of n is 21 . Now, break 21 into its factor, 21=7×3 ,7,3=1 If m>2 and n>2 then the…

Q: Q1. Find the multiplicative inverse of 29 modulo 1137. Write down all steps of your computa- tion.…

A: For any two arbitrary integers a and n, if the existence of another integer b is established so that…

Q: Consider the Baby-Step, Giant Step Algorithm to solve 2* = 11 mod 13. The least common element…

A: The given algorithm is, 2x≡11 mod 13  r=3 and k=5  Let a=2 and b=11

Q: Show that if p is an odd prime, then 2(p - 3)!= -1 (mod p) Prove that if n has an odd number of…

A:

Q: 2-) Find all in tegers x such thet 5x = 12 (mod 7) 10x = 4 (mod 1) 7x E 16 (mod 10d ond 1occ < x <…

A: Given: 5x≡12(mod 7) 10x≡4(mod11) 7x≡16(mod 10) and 1000≤x≤3000

Q: Find the inverse of the congruence 6x = 1 (mod 29) using Euclidean algorithm. (Show your work)

A: The given problem is to solve the given modular equation by find the inverse of congruence using the…

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

A: Given a =463 and  n=953 We need to find x such that                   463952 ≡ x (mod 953).

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.

A:

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

A:

Q: Use 672 = -1 mod 449 and Hermites' algorithm to find a pair of positive integers s and t such that…

A: Given :          s2 +t2 = 449672 = -1 mod449

Q: Compute 17^7121 (mod 11).

A:

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

A:

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

A:

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

A:

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

A:

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

A:

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 natural numbers b with 100 < b < 200 such that: 5211 b (mod 35)

A:

Q: Apply the Extended Euclidian Algorithm to find 1/21 modulo 59 with a table

A: We will apply Extended Euclidian Algorithm to find 121 modulo 59 i.e., multiplicative inverse of 21…

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

A:

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: Let a, k and n be positive integers. Describe a fast algorithm to compute ak MOD n. Illustrate your…

A:

Q: Compute 21200 mod 27.

A: We see,         21 ≡ (-6)  mod 27     =&gt; 212 ≡ (-6)2  mod 27                 ≡ 36  mod 27…

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: Rewrite 17x=3 (mod 53) as linear combination. b) Using Euclidean Algorithm to find the gcd(17, 53).…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: we have 1529 MOD 83 = 2 Find the inverse to 2 by using the Euclid's algorithm

A: The given expression is 1529 mod 83=2. We have to find the inverse of 2 by using Euclid's algorithm.

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

A:

Q: Compute 21200 mod 27.

A:

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: (a) Write 13 as a sum of distinct powers of 2. (b) Use the fast factoring algorithm to compute 213…

A: 13 can be written as a sum of distinct powers  of 2. 2^3=8 2^2=4 2^0=1  2^3 + 2^2 +2^0

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

A:

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: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: Use Chinese theorem to find all positive integers that makes this congruence true:        x = 2 mod…

A: