3. Find a div m (meaning find quotient and remainder) and a mod m when: (a) a = 10299, m = 999 (b) a = 123456, m = 1001 (c) a = −111, m = 99 (d) a = −1000, m = 101
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...
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?
Find the additive inverse of the following value mod m. 16. 7 mod 9 17. 4 mod 12 18. 63 mod 30 19. 222 mod 3
Find the smallest positive inverse of 10 (mod 17)
Problem 3. Use the Chinese Remainder Theorem to find all congruence classes that satisfy x2 = 1 mod 77.
8. Find the remainder when 21000000 is divided by 17.
4. Find each of these values: (a) (177 mod 31 + 270 mod 31) mod 3 (b) (177 mod 31 · 270 mod 31) mod 31
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,