Apply properties of congruence to make computation in modulo n feasible. Find the unique r such that a is logically equivalent to r (mod n ) with 0 less than or equal to r and r less than n.  a) (25)2022(27)2022 ≡ r (mod 5) b)(1234567891237456789)2(1234567891)2022 ≡ r(mod 10) c) (3)2022 (9)2022 ≡ r (mod 27)

Q: 2. Euclidean Algorithm Use the Euclidean algorithm to find gcd(12345, 678).

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Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: Express each nonnegative integer a less than 15 as a pair (a mod 3, a mod 5).

A: The values of a could be, 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: 1. Find all solutions (a) 2x + 3 = 7 (b) 2x + 3 = 7 to the following congruences.. (mod 9) (mod 20)

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Q: Compute or show that it does not exist. (a) 149-¹ (mod 200); (b) 95-¹ (mod 323).

A: Please see the below picture for detailed solution.

Q: ) Find all possible values of m: 53 = -7 mod (m)

A: m is positive greater than 1.

Q: 2) Find 281171 + 171281 (mod 12)

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Q: If a ≡ b (mod n1) and a ≡ b (mod n2), must it be true that a ≡ b (mod n1n2)? What if gcd(n1, n2) =…

A: If a≡b(modn1) and a≡b(modn2), it is not necessarily true that a≡b(modn1n2). The statement is true…

Q: Perform the following congruence computations. Make sure that the number you enter is 20 and ≤N-1,…

A: (a) 5946+7925≡        mod 45⇒13871≡        mod 454530813871-13500371-360011∴5946+7925≡  11      mod…

Q: 2. Encrypt the message STOP POLLUTION by translat- ing the letters into numbers, applying the given…

A: let's encrypt the message "STOP POLLUTION" using the given encryption functions one by one.For the…

Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

Q: 122 is congruent to 5 modulo17 ?

A: To check whether -122 is congruent to 5 modulo 17 or not

Q: 17!=1(mod19). Select one: True False

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Q: Show that y = x (mod p) leads to -y = -x (mod p).

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Q: Suppose that a and b are integers, c b = 3 (mod 19). Find the integer c that a) с%3D 13а (mod 19).…

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Q: State whether True or False: If A and B are integers and A ≡ 3 (mod 3) and B ≡ 2 (mod 6) then A+B ≡…

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Q: Assume all variable as positive integers. Given and a-2n+2 (mod 7) 000 b = 3n+1 (mod 7) Which of the…

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Q: 237 (mod 10)

A: We use the definition of a (mod b) to find the value of 237 (mod 10).