13. Suppose that a and b are integers, a = 4 (mod 13), and b = 9 (mod 13). Find the integer c with 0 ≤ c ≤ 12 such that a) c = 9a (mod 13). b) c = 11b (mod 13). c) c = a + b (mod 13).

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A: To explain why Christmas and new year's day always fall on the same day of the week.

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A: We have to prove or disprove the following statement For all positive integer n and for all integer…

Q: 7. List five numbers that belong to the 5 (mod 6) numbers. Alternate phrasing, list five numbers…

A:

Q: 9. Let a and b be integers. Prove that if a =7 (mod 8) and b = 3 (mod 8), then: (a) a + b = 2 (mod…

A: Your questions solution below in images

Q: 2. What is the least residue of 610 C 512 10.4512 (mad 1112

A: Euler's theorem is used

Q: Find n ∈ {1, 2, 3, . . . , 28} that satisfies 26! ≡ n (mod 29).

A: Given ,              26! ≡ n mod(29)  Then we have to find the  value of n∈  1 , 2 , 3 , .........,…

Q: (c) For each integer a, if a 4 (mod 8), then a 2 (mod 8).

A:

Q: л There

A: We prove our result by using definition of mod.

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: If x = 2 (mod 5) and y = 4 (mod 5) then 2x – 3y =? (mod 5)

A:

Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

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Q: Suppose that a and b are integers. a ≡ 7 (mod 13) and b ≡ 9 (mod 13). Find the integer c such that 0…

A: The given problem is to find the modular values of c for the given conditions with given integers a…

Q: a) If a| (21b−c) and 6c≅3b(mod2a), then a| 41c b) If a≅b(modc) and d|c then ?≅?(mod?) c) ∴(d|…

A: this is a question of modular arithmatic.

Q: 1. If a, b are integers and 3a or 3 b, show that 3 (a² + b2²).

A: If a, b are integers

Q: Is it true that 90 ≡ 64 (mod 13)?

A: As given equation 90≅64(mod 13) is False.

Q: 7. Prove that every integer is congruent modulo 9 to a unique integer T such that -4≤r ≤ 4.

A:

Q: *1. What is the least residue of 56 (mod 7) 58 (mod 7) 19458 (mod 7)?

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

Q: 5. Prove that every integer is congruent modulo 9 to one of the even integers 0, 2, 4, 6, ..., 16.

A: To prove that every integer is congruent modulo 9 to one of the even integers

Q: 9. Prove that if n = 3 (mod 4), then the equation x² + y² = n has no integer solution.

A:

Q: 16. Prove that an integer (a,-1an-2 · . . ao)10 is divisible by 11 if and only if ao + a2 + a4 + . .…

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Q: Need help with True or False for the following... (I will put what I believe As (T) or (F)) 1. For…

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