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

Q: Find the smallest positive integer k such that 3k = 1 (mod 19).

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Q: Use Fermat's Little Theorem to find each value. Remember, x-1) =1 (mod p). Write your answer in…

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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: 4444 is divided by a 1.19) find the remainder when 4444 * 2?=-1 (mod a).

A: 1.19 We have 4444  ≡  7 (mod 9). So, 44444444 ≡ 7 4444 (mod 9) Now,       73 ≡1 (mod 9) Or,…

Q: Find all solutions for x, up to congruence. x ≡ 1 mod 4 and x ≡ 7 mod 13.

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

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Q: (a) Prove that 2 and 3. are both primitive roots modulo 5.

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Q: Suppose that a is an odd integer and (a, 91) = 1. Prove that a12 ≡ 1 (mod 1456).

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

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Q: A graphing calculator is recommended. Use technology to find the volume of the solid obtained by…

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Q: (c) Does the polynomial x² - 27 = 0 (mod 1271) have any roots? Note that 1271 = 31 - * 41 is not a…

A: We use the Chinese Remainder Theorem to find the solutions to the congruence x2-27≡0(mod1271). Now…

Q: (d) (e) (f) Assume that a 15 = 1 Find the value of ord41a. mod 41 and a #1 mod 41. How many…

A: As per the question we have to find : ord41(a) , where a15 ≡ 1 mod 41 but a ≠ 1 mod 41 total number…

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

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Q: For each part, use the data provided to find values of a and b satisfying a2 ≡ b2 (mod N), and then…

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Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

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

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Q: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)

Q: Find all possible positive values of m: 47 = m mod (5)

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Q: Find all solutions to the congruence13x−3 = 4 (mod 26). If none exists, explain why.Your…

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