Let 1Sasb be two positive integers. How can I prove or disprove that there exists a 1sks b-1 such that ak=1 (mod b)

Q: Prove that if a and b are integers with a 0 and where + b - a = 0, x is a positive integer such that…

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Q: Prove that x113+x-1≡0(mod 113) has no solutions in x.

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Q: Find all x (mod 1999) such that x³ = 2 (mod 1999).

A: We use primitive roots to solve this problem.

Q: Find the matrix A of the linear transformation from R to R' given by 5 T 6 x1 + -1 x2. %3D -8 7 A =

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Q: How do you fiNd all positive integers m for which the following congruence is true. 100 ≡ 50(mod m)

A: We know if a≡b( mod m) then a-b = mk where k is integer. Given,      100 ≡ 50 ( mod m ) We have to…

Q: . Determine whether or not x³ + 2x + 1 and x4 + 3x - 2 are congruent mod (x² + 2x + 2) in Q[x].

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Q: Give a definition of an inverse of a number a modulo m.

A: if the two integer a and m modulo then  let x is an integer value so the x is inverse of an only if…

Q: Find a positive integer less than 6 for x that sat 2x + 340 = 1703 (mod 5).

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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 four solutions to x2 =6 (mod 19 x×23) with 1<x< 437.

A: Solution:-

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

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Q: Test2

A: We know that2198=2128×264×24×22Also we have22≡4(mod 397)24≡16(mod 397)28≡256(mod…

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

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Q: For integers a and b, prove that (a + b) 3 a3 + b3 (mod 3).

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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: Let a € R, and define vectors u₁, U2, U3 € R³ in terms of a by E] 2 U₂ = a U₁ = 3a +3 2a +3 a² + 3a]…

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Q: 2. Let = = 4cjs49°, = 2/2cjs92°, and , = cjs (-86°). Evaluate the expression ち+(-)

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Q: Determine a positive integer x such that 57x +7 = 78 (mod53).

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Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: (b) Find 95! (mod 96).

A: We have to find the value of 95!(mod 96). Since 95!=95*94*93*…*8*7*6*5*4*3*2*1.

Q: Use a Proof by Contradiction to prove that there is no integer a such that a congruent 5 (mod 14)…

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Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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Q: Is 3 a primitive root of 43? Use your answer to determine all positive integers less than 43 whose…

A: To determine whether 3 is a primitive root of 43, we need to check if it generates all the positive…

Q: (b) Compute 9-¹ (mod 19) by listing the multiples. Show your work.

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Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

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Q: Show that there are exactly (p−1)/2 nonzero squares mod p.

A: To prove that there are exactly p-12 nonzero squares mod p.

Q: Let a be an odd integer. If a 3 we have (2ª) a 2 = 1 (mod 2°),

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Q: x² + 4x = 10(mod11) (mod11) X=

A: Given that x2+4x≡10 (mod 11)x2+4x+4≡10+4 (mod 11)x2+4x+4≡14 (mod 11)x2+4x)(x+2)2≡14+11 (mod…

Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: x² + x = 13(mod17) X= * (mod17)

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Q: Show that there are no integers a, b such that a² + 2ab+26² = 100000000003. Hint. Use mod 4…

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Q: Establish that if a is an odd integer, then 2" а = 1 (mod 2" +2) for any n 2 1.

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Q: Let a and b be relatively prime positive integers. Show that a(b) + b(a) = 1(mod ab).

A: Solution:

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

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Q: (a) Prove that 318 = q (mod 19).

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Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

Q: 1. Prove that an integer n can not be represented as a difference of two squares if and only if n 2…

A: The given problem is to prove that integer n can't be expressed as difference of two squares if and…

Q: If n is an integer, then (2n² + 3n)# 3(mod5).

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Q: Prove that for any integer x, if x mod 3 does not divide 0, then (x2 −3x + 1) mod 3 = 2.

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Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

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