1. For each solution. congruence determine whether there is a J X² = -1 (mod 397) (397 is prime) CC x² + 8x + 14 = 0 (mod 37) iii) x² = 785 (mod 787) (787 is prime)

Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: Find all solutions of the congruence 8x ≡ 4 (mod 12)

A: Step 1: Divide by the greatest common divisor (GCD)The GCD of 8 and 12 is 4. We can divide both…

Q: Find all values of x (using congruent notation) for 35x ≅ 3 (mod 89)

A:

Q: Find all solutions in {0, 1,2, ..., 12} to x26 + 15x13 = 12 mod 13.

A:

Q: Find the common solutions to the congruences 2x 3x 4x 1 (mod 5) 9 (mod 6) 1 (mod 7).

A: Given : 2x≡1 (mod 5)3x≡9 (mod 6)4x≡1 (mod 7) To find : The common solutions to the congruences. For…

Q: Solve the system of congruences 2x ≡ 1 (mod 5) 3x ≡ 9 (mod 6) 4x ≡ 1 (mod 7) 5x ≡ 9 (mod 11) Hint:…

A: We can solve this system of congruences using the Chinese Remainder Theorem. First, let's write each…

Q: 1. For each solution: determine weather there is a a congruence a x² = -1 (mod 397) b.) x²+8x+14=0…

A: 1a) To determine whether the congruence equation x2 ≡ -1 (mod 397) has a solution, we need to check…

Q: Find all solutions for x, up to congruence. (b) x ≡ 11 mod 142 and x ≡ 25 mod 86.

A: Given: (b) x≡11 mod 142 and x≡25 mod 86 To find: All solutions for x, up to convergence.

Q: SorvE THモ CONGROEモNしE 626 X Rエ mod5

A: As 5 is an odd number, by fermat's theorem, we have             x5-1 = 1 (mod 5)           => x4…

Q: Solve the following congruences. Make sure that the number you enter is ≥ 0 and ≤ N-1, where N is…

A:

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: Find a number a such that 70a = 1 mod 17.

A:

Q: 3. Solve the following congruences. † (a) 10x = 5 (mod 21) (b) 10x = 5 (mod 15) † (c) 10x = 4 (mod…

A:

Q: The remainder when aa is divided by 77 is 22 and the remainder when bb is divided by 77 is 55. (So a…

A: Given:          a≡2 mod(7)  and b≡5 mod(7)

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

A:

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 ≡…

A:

Q: Given that 3599 = 59-61, determine whether x² = 581 (mod 3599) has solutions.

A:

Q: Show that if x mod 8 = 1, then (3x −5) mod 6 = 4.

A: Solution