Section Number Points Comments
2.4 6 4
10(c),(f),(h) 3
12(a),(b) 2
14 5 Don't just give the answer. Prove it! Hint: Think in
terms of prime factors. (There's a reason that the
problem is in this chapter ...)
16 3
20 5
28(a) 2
30(a) 2
44 3
46 3
2.5 20 4
22(c),(d) 6
28 5 Hint: what is the congruence class of 10 mod 11
Extra problems:
- 1. [5 points] Do problem 0.2, 33 in DAM3 (DAM2: 0.4, 24), then prove (by induction) that
your formula for f(n) in part (b) is correct.
- 2. [8 points] Let S be the smallest set such that has the following two properties:
- S1. 1 is in S, and
- S2. if x is in S then x+2 is in S.
Define On as inductively as follows:
- O1 = {1}
- O(n+1) = On ∪ {2n+1}.
Let O = ∪ On.
- (a) Prove that by induction that On = {1,3, ..., 2n-1}.
Note that it follows that O is the set of odd numbers.
- (b) Prove that O = S. (Recall that this means you have to prove
that S is a subset of; O and O is a subset of S. To show
that S is a subset of O, use the fact that S is the smallest
set satisfying S1 and S2. To show that O is a subset of S, prove by
induction that On is a subset of S.)