
Find all integers x, y, 0 < x, y < n, that satisfy each of the following pairs of congruences. If no solutions exist, explain why. (a) x + 5y = 3(mod n), and 4x + y = 1(mod n), for n = 8. (b) 7x + 2y = 3(mod n), and 9x + 4y = 6(mod n), for n=5.
Find the smallest positive solution and the general
solution to the system x ≡ 1 (mod 3), x ≡ 2 (mod 5) and x ≡ 3 (mod
7).
Exercise 2 (5 points Find the smallest positive solution and the general solution to the system ΧΞ2 (mod 5) and r Ξ 3 (mod 7). 1 (mod 3),
Question 13 (0.5 points) For all positive integers a and b, if al0 = 1 (mod b) then a = 1 (mod b). True False Question 14 (0.5 points) For all positive integers an, az, mi and m2, if mı #m2 then the system of linear congru- ences x =ai (mod mi) x = a2 (mod m2) admits at most one solution modulo mim2. True O False Question 15 (0.5 points) For all positive integers a and b, if a|b2...
Problem 1 Use the Chinese remainder theorem, find all integers x such that: (20 pts) x = 1 (mod 5) x = 2 (mod 7) x = 3 (mod 9) x = 4 (mod 11)
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,
find all integers such that (x^86) is equivalent to (6 mod 29) . plese explain each step in detail
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...
27) Solve each equation for x. a) +4 0, (mod 9) b) 4x+8 5,(mod 11) Xr
Solve the following for x 4x ≡ 7 (mod 19) 6x ≡ 8 (mod 31) 9x ≡ 7 (mod 16)
Find all the integers x which are the solutions to the following congruences. x^2 is equivalent to 2 mod 17