Determine whether each of the following congruences is solvable, and find the number of incongruent solutions of the each congruence. (i) x² = -471 mod 4. 136. 238.4131 (ii) x² = -471 mod 28. 2312.61¹1 (iii) x² = -471 mod 3.4115 (iv) x² = -471 mod 35 - 23²

2. Solve the simultaneous congruences x = 4 (mod 5) and 2x = 3 (mod 7)

Find all solutions mod 31 to the congruence x° = -5 (mod 31).

1+n 3n+1 For each natural number n, Let - Find UA, and 4. n n EN EN If R={(x, y) & RxR: x² = 3 +5y}, S={(x,y) & R×R: y³ = 3x²+5}, Then find RoS and SoR Solve the simultaneous congruences: x = 3 (mod 2) x = 5 (mod 7) (x = 11 (mod 13)