EXERCISES 1. For n = 15, 18, 22, 30, and 35, find all positive integers less than n and relatively prime to n.

Answer :As per the question,Note:''A positive integer is any integer that is greater than zero (0)''.  Also, a positive integer is any integer that is a member of the set of natural numbers, i.e., counting numbers; therefore, the positive integers are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 34, 25, 26, 27, 28, 29, 30, 31, 32, 33, ...“Relatively Prime” (also called “co-prime”) numbers are numbers whose HCF is 1.  Any consecutive positive integers are co-prime (e.g.: 42 and 43)Now,First case: If n = 15Positive integers (less than n = 15) are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14.Relatively prime are : (1 and 15), (2 and 15), (4 and 15), (7 and 15), (8 and 15),  (11 and 15), (13 and 15) and (14 and 15).Second case: If n = 18Positive integers (less than n = 18) are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17.Relatively prime are : (1 and 18), (5 and 18), (7 and 18), (11 and 18), (13 and 18) and (17 and 18).Third case: If n = 22Positive integers (less than n = 22) are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 and 21.Relatively prime are : (1 and 22), (3 and 22), (5 and 22), (7 and 22), (9 and 22),  (13 and 22), (15 and 22),  (17 and 22),  (19 and 22) and (21 and 22).Fourth case: If n = 30Positive integers (less than n = 30) are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 and 29.Relatively prime are : (1 and 30), (7 and 30), (11 and 30), (13 and 30), (17 and 30), (19 and 30), (23 and 30) and (29 and 30).Fifth case: If n = 35Positive integers (less than n = 35) are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33 and 34.Relatively prime are : (1 and 35), (2 and 35), (3 and 35), (4 and 35), (6 and 35), (8 and 35), (9 and 35), (11 and 35), (12 and 35), (13 and 35), (16 and 35), (17 and 35), (18 and 35), (19 and 35), (22 and 35), (23 and 35), (24 and 35), (26 and 35), (27 and 35), (29 and 35), (31 and 35), (32 and 35), (33 and 35) and (34 and 35).