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the second part of the question can be solved by the chineses
remainder theorem.
Problem 4...
please complete exercises 10.4, 10.5, 10.6, 10.7 and 10.9, thank you so much! (I dont understand...
please
complete exercises 10.4, 10.5, 10.6, 10.7 and 10.9, thank you so
much! (I dont understand your comment what is qs 3.6?)
10.4 Exercise. Show that the algorithm descrihed in Question 3.6 for com puting a (mod n) is a polynomial time algorithm in the number of digits in r In the next scrics of problems you will cxplore the usc of this opcration as a means of testing for primality by starting with a familiar theorem. Theorem (Fermat's Little...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
Problem 1 Use the Chinese remainder theorem, find all integers x such that: (20 pts) x...
Problem 1 Use the Chinese remainder theorem, find all integers x such that: (20 pts) x = 1 (mod 5) r = 2 (mod 7) x = 3 (mod 9) I= 4 mod 11) Answer,
9. Use the construction in the proof of the Chinese remainder theorem to find a solution to the system of congruences X 1 mod 2 x 2 mod 3 x 3 mod 5 x 4 mod 11 10. Use Fermats little theorem to fin...
9. Use the construction in the proof of the Chinese remainder theorem to find a solution to the system of congruences X 1 mod 2 x 2 mod 3 x 3 mod 5 x 4 mod 11 10. Use Fermats little theorem to find 712 mod 13 11. What sequence of pseudorandom numbers is generated using the linear congruential generator Xn+1 (4xn + 1) mod 7 with seed xo 3?
9. Use the construction in the proof of the Chinese...
Please help me with understandable solutions for question 6(a), 7, 8 and 10. ( Use Chinese...
Please help me with understandable solutions for question 6(a), 7,
8 and 10. ( Use Chinese remainder theorem where applicable).
78 CHAPTER 5. THE CHINESE REMAINDER THEOREM 6. (a) Let m mi,m2 Then r a (mod mi), ag (mod m2) can be solved if and only if (m, m2) | a1-a2. The solution, when it exists, is unique modulo m. (b) Using part (a) prove the Chinese remainder theorem by induction. 7. There is a number. It has no remainder...
3. (16 points) Solve the system of linear congruences using the Chinese Remainder Theorem. 4 (mod...
3. (16 points) Solve the system of linear congruences using the Chinese Remainder Theorem. 4 (mod 11) a 11 (mod 12) x=0 (mod 13) b. (6 pts) Find the inverses n (mod 11), n21 (mod 12), and nz1 (mod 13). Using these ingredients find the common solution a (mod N) to the system. c. (4 pts) 4. (8 points) What is 1!+ 23+50! congruent to modulo 14?
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers....
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers. Let s,t E Z be such that sm+tn The Chinese Remaindering Theorem states that for every a, b E Z there exists c E Z such that r a mod m (Va E Z) b mod nmod mn (3) where a convenient c is given by 1. Prove that the above c satisfies both ca mod m and cb mod n 2. LetxEZ. Prove...
please do 7.19 7.20 and 7.21 7.19 Theorem (Quadratic Reciprocity Theorem and q be odd primes,...
please do 7.19 7.20 and
7.21
7.19 Theorem (Quadratic Reciprocity Theorem and q be odd primes, then Reciprocity Part). Let p (e)99 (mod 4) if p (mod 4) or q1 i p 3 (mod 4). (i)) (llint: Iry to use the techniquets used in the case of Putting together all our insights, the Law of Quadratic Reciprocity. we can write one theorem that we call Theorem (Iaw of Quadratic Reciprocity). Let p and q be odd primes, then if p...
D Question 5 7 pts Use division and the Remainder Theorem to find the value of...
D Question 5 7 pts Use division and the Remainder Theorem to find the value of P (i). Where P(x) = 624 – 2x2 + 4.
Number theory: Part C and Part D please! QUADRA range's Four-Square Theorem) If n is a natural be expressed as t...
Number theory: Part C and Part D please!
QUADRA range's Four-Square Theorem) If n is a natural be expressed as the sum of four squares. insmber, then n cam be expressed tice Λ in 4-space is a set of the form t(x,y, z, w). M:x,y,z, w Z) matrix of nonzero determinant. The covolume re M is a 4-by-4 no is defined to be the absolute value of Det M such a lattice, of covolume V, and let S be the...
please
complete exercises 10.4, 10.5, 10.6, 10.7 and 10.9, thank you so
much! (I dont understand your comment what is qs 3.6?)
10.4 Exercise. Show that the algorithm descrihed in Question 3.6 for com puting a (mod n) is a polynomial time algorithm in the number of digits in r In the next scrics of problems you will cxplore the usc of this opcration as a means of testing for primality by starting with a familiar theorem. Theorem (Fermat's Little...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
Problem 1 Use the Chinese remainder theorem, find all integers x such that: (20 pts) x = 1 (mod 5) r = 2 (mod 7) x = 3 (mod 9) I= 4 mod 11) Answer,
9. Use the construction in the proof of the Chinese remainder theorem to find a solution to the system of congruences X 1 mod 2 x 2 mod 3 x 3 mod 5 x 4 mod 11 10. Use Fermats little theorem to find 712 mod 13 11. What sequence of pseudorandom numbers is generated using the linear congruential generator Xn+1 (4xn + 1) mod 7 with seed xo 3?
9. Use the construction in the proof of the Chinese...
Please help me with understandable solutions for question 6(a), 7,
8 and 10. ( Use Chinese remainder theorem where applicable).
78 CHAPTER 5. THE CHINESE REMAINDER THEOREM 6. (a) Let m mi,m2 Then r a (mod mi), ag (mod m2) can be solved if and only if (m, m2) | a1-a2. The solution, when it exists, is unique modulo m. (b) Using part (a) prove the Chinese remainder theorem by induction. 7. There is a number. It has no remainder...
3. (16 points) Solve the system of linear congruences using the Chinese Remainder Theorem. 4 (mod 11) a 11 (mod 12) x=0 (mod 13) b. (6 pts) Find the inverses n (mod 11), n21 (mod 12), and nz1 (mod 13). Using these ingredients find the common solution a (mod N) to the system. c. (4 pts) 4. (8 points) What is 1!+ 23+50! congruent to modulo 14?
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers. Let s,t E Z be such that sm+tn The Chinese Remaindering Theorem states that for every a, b E Z there exists c E Z such that r a mod m (Va E Z) b mod nmod mn (3) where a convenient c is given by 1. Prove that the above c satisfies both ca mod m and cb mod n 2. LetxEZ. Prove...
please do 7.19 7.20 and
7.21
7.19 Theorem (Quadratic Reciprocity Theorem and q be odd primes, then Reciprocity Part). Let p (e)99 (mod 4) if p (mod 4) or q1 i p 3 (mod 4). (i)) (llint: Iry to use the techniquets used in the case of Putting together all our insights, the Law of Quadratic Reciprocity. we can write one theorem that we call Theorem (Iaw of Quadratic Reciprocity). Let p and q be odd primes, then if p...
D Question 5 7 pts Use division and the Remainder Theorem to find the value of P (i). Where P(x) = 624 – 2x2 + 4.
Number theory: Part C and Part D please!
QUADRA range's Four-Square Theorem) If n is a natural be expressed as the sum of four squares. insmber, then n cam be expressed tice Λ in 4-space is a set of the form t(x,y, z, w). M:x,y,z, w Z) matrix of nonzero determinant. The covolume re M is a 4-by-4 no is defined to be the absolute value of Det M such a lattice, of covolume V, and let S be the...
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