I. Solve for x. All final answers must be in Znwhere nis the modulus.  (a) x≡3(15) + 45 mod 18 (b) x≡−12 mod 13 (c) 6xmod 11 ≡1 (d) x+ 2 mod 9 ≡0 II. Write down the congruence classes formed, modulo 4.  III. Find a multiplicative inverse of 3 modulo 6, if there are any.

Q: The statement a ≡ b (mod n) is called a congruence.

A: Given that  The statement a ≡ b (mod n) is called a congruence.

Q: 3. Use the table of indices modulo 17 below to solve the following problems. 1 2 3 4 5 6 7 8 9 10 11…

A: Given: The table of indices modulo 17.To find: a) 8x6≡2mod17              b) 8x4≡15mod17…

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Q: 1. Show that k1601 = k (mod 17) for any k.

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Q: 2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution…

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Q: Solve each of the following congruences. (a) 6x + 3 = 4 mod 10 (b) 6x + 3 = 1 mod 10 (c) x² = 1 mod…

A: In congurent modulo n, ax≡bmod n has exactly 'd' solutions if 'd' divide 'b'where d=gcda,n otherwise…

Q: Suppose that a and b are integers such that a = 45 (mod 71) and b = 53 (mod 71). Find an integer c…

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Q: Can you please show me step by step for this math that makes me undertandable. Q: Determine which…

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Q: C1. A. 32 B. 234 C. 238 The multiplicative inverse of 177 mod 260 is D. 37 E. does not exist F. none…

A: To find the multiplicative inverse of 177(mod260).

Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: 3 Р 1 ? What if 3 is replaced by = 2. For which primes p is -3, 5, -5, 6, or -6 ? Any patterns? Is…

A: The symbol  is the Legendre symbol, used in number theory to denote whether a number is a quadratic…

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Q: 5. Let p be an odd prime, and suppose that ged(a, p) = 1. Show that the congruence z = a (mod p*)…

A: p is an odd prime and gcd(a,p)=1We have to show the congruence x2=a mod pk has exactly 1+ap…

Q: - 1)/2 many integers: n, 2n, P=¹n. ∙n. 2

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Q: If x98 = 1(mod 97) then possible value of x is of (a) 97 (b) 194 (c) 90 (d) 291 (е) None of these O…

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Q: 1. A+B (mod 173) 2. C-D (mod 432) 3. 23 x E (mod F) 4. G+5 (mod 7) АВМ 1А A B C E F G Hannah 7550…

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Q: Determine the cycle or lifetime (N) of the following congruential generator. Xi+1 = (13Xi + 9) mod…

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Q: (3) Solve for x (give your answer as an integer between 0 and 7: 14 · 51 ≡ x(mod7)

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Q: Informal Exercise 20. Observe that 4| 102. Show that to find Rem(n, 4), you just need to replace n e…

A: We need to find the remainder when n is divided by 4 .

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Q: 9. Find all integers x, 0 < x < n, satisfying each of the fol- lowing congruences mod n. If no such…

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Q: Determine the cube root of (3 + j4). ound-off the modulus to 3 decimal places, the angle to the…

A: Cube root of (3+4j)

Q: (a) Prove the following: i. 11 + (6 x 4") - 22n = 6 mod 7, 1 is divisible by 131, ii. 8260 iii.…

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Q: Solve for x. All final answers must be in Zn where n is the modulus. 6x mod 11 ≡1

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Q: 2) Solve for x (give your answer as an integer between 0 and 9): 51 + 27 ≡ x(mod10)

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Q: Q4 (a) Solve for x if 25x = 15(mod 29). (b) Determine the smallest positive integer so that when…

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Q: 38. It is known that 14 is a primitive root for the prime p = 30000001. Let 14⁹000000 (mod p). (The…

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Q: y²=x³-5x+3 (mod5). solve for all x and y, when x is at least 0, and y is at most 4

A: Using congruence and modular arithmetic we solve the given equation. We substitute the value of x…

Q: Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can…

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Q: Define the following operation on Zn: { [1] if a = b (mod 5) [0] if a zb (mod 5) (a] * (b] = (a) Is…

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Q: 11.5. For each part, find an x that solves the given simultaneous congruences. (a) x = 3 (mod 7) and…

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Q: True or False. Briefly explain why.  By Fermat’s Little Theorem, 6 is the smallest positive integer…

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Q: VI. Determine four values of x that will satisfy the following. 16. x = 4 mod 5 & x = 3 mod 7

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Q: Q7. Solve the following set of simultaneous congruences. 6x = 1(mod 5) 5x = 3(mod 11) 2x = 5(mod 13)

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Q: Find the modulus and argument of 5 - 7i.

A: Since the modulus of a complex number a+ib is |a+ib|=a2+b2. So, the modulus of 5-7i is…

Q: Exercise 6.3.4. Find all solutions modulo 10 to x² + 3x = 0 (mod 10). There are two distinct…

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Q: QUESTION 5 Consider the linear combination 1 = 13(3) + 19(-2) then, a. The multiplicative inverse of…

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Q: 22. Which of the following integers are the possible one's digits of a fourth power of any 0, 1, 5…

A: # as per the guidelines we are entitled to solve one question at a time, please resubmit the other…