Solve pls

11 12. 13. 14. 15. 16. 17. 18. Let d = (826, 1890). Use Euclid's algorithm to compute d and then express (1 as a linear combination of 826 and 1890. Show that 0" lb\" implies a | 1:. Determine all solutions in the positive integers of the following Diophantine equation 172x + 20y = 100. (Ans. (5, 7)) Find the remainder when 4165 is divided by 7. Give an example to show that 612 = 122 (mod 11) need not imply that a 5 13 (mod n). Prove that if ab a cd (mod n) and b = d (mod n) with g.c.d.(b, n) = 1, then a = c (mod 13). If a = b(mod 111), and a = C(mod n2), prove that b = 0 (mod n) where n = (121, 122). Which of the following congruences hold (1') 12, 345, 678, 987, 654, 321 =0(mod 12, 345, 678) (a) 12, 345, 678, 937, 654, 321 :0 (mod 12, 345, 679) 19. Solve the system of congruences (1') x a 5 (mod 7), x -=- 3 (mod 5), x a 2 (mod 8), x a 2 (mod 3) (if) x a 4 (mod 5), x a 6 (mod 3), x a 2 (mod 3) 20. 21 . Prove that if '0' 41%, lb I