(1) Find all solutions of the congruences (a) 20;}: E 4 (mod 30) (b) 20:: E 30 (mod 4) (c) 353.1: E 254 (mod 400) (d) 573: E 87 (mod 105) (e) 641? E 83 (mod 105) (f) 589:: E 209 (mod 817) (g) 491? E 5000 (mod 000) (2) Find an inverse modulo 17 for each of the following integers. (a) 2 (b) 3 (C) 5 (d) 11 (3) Find the solutions of each of the following systems of linear congruences. (a) 23: + 3y 5 (mod 11) x + 5;; 6 (mod 11) 0?) 4x + y E 5 (mod 11) 3: + 33; E 4 (mod 11) (4) What is the last digit in the ordinary decimal representation of 2400? Justify your answer. (5) Find all the solutions of each of the following system of linear congruences. (a) 1: E 4 (mod 11),;12 E 3 (mod 17) (b) 1:: 1 (mod 2), :5 E 2 (mod 3)3: E 3 (mod 5) (c) 1? E 0 (mod 2) 5r. 0 (mod 3) 3: E 1 (mod 5), J: E 6 (mod 7) (d) 1: E 2 (mod 11) 35': 3 (mod 12) r E 4 (mod 13), .'I' E 5 (mod 17) 12E 6 (mod 19). (5) Solve the following ancient Indian problem: If eggs are removed from a basket 2, 3, 4, 5, and 6 at a time, there remain, respectively 1, 2, 3, 4, and 5 eggs. But if the eggs are removed 7 at a time, no eggs remain. What is the least number of eggs that could have been in the basket? (7) How many incongruent solutions are there to the congruence 3351126 E 0 (mod 144)