Section 2.3

• 16. a) 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14. b) Proper divisors are in two batches: 1, 2, 4, ..., 2^(p-1) and the same sequence all multiplied by 2^p - 1 but not the final term of 2^(p-1) (2^p - 1). Each is a geometric series. The sum of the first is 2^p - 1 and of the second is (2^p - 1)(2^(p-1)-1). Add them together to get 2^(p-1) (2^p - 1).
• 22. If n is prime, then phi(n) = n-1. If n is not prime, then phi(n) < n-1!
• 30. 2^4 3^4 5 7^11
• 34. b = a + sm, so bc = ac + s(mc), so ac = bc (mod mc).
• 38. a) 58, b) 60 c) 52 d) 3. You need a calculator or program that can work with at least 13 digits of accuracy.
• 40. 3 6 4 3 6 4 3 ...
• 42.