# Homework

I will assign problems for class discussion every day. You should do the listed problems before the corresponding class so that you are prepared to discuss them.

Periodically I will select a subset of the problems I have assigned to be handed in. The selected problems will be announced with a relatively short lead time. Thus it is in your interest to stay current on the homework.

Chapter 1 (page 20ff)

• Aug 26:
• Aug 28: 1, 3, 10, 14, 22
• Aug 30: 23, 24, 27, 57, 60, 80
• Sep 4: 40, 42, 47, 73, 74
• Sep 6: 62, 64, 68, 69, 84a
• Sep 9: 77, 83, 84b

Chapter 2

• Sep 11: 1,3,8,10, 99
• Sep 13: 14, 19, 27, 28, 43, 46, 75
• Sep 16: 51, 52, 58, 73, 76, 77, 94
• Sep 18: 81, 84, 93, 101

Chapter 3

• Oct 2: 1,2,3,6,7
• Oct 4: 8,11, 56, 62, 63
• Oct 7: 22, 24, 26, 31, 48, 67, 68
• Oct 9: 65, 66, 98, 99.
• Oct 11: 71, 101

Chapter 4

• Oct 14: 1, 5, 7, 10, 11, 49
• Oct 16: 15, 16, 21, 24, 25, 52
• Oct 18: 28, 31, 32, 43, 56, 80
• Oct 21: 59, 60, 61, 62, 63, 82

Chapter 5 and The Book Of Proof

• Oct 21: Hammack pg. 264: 1,3,8,9
• Oct 23: Hammack pg. 264: 4,5,7,10,11
• Oct 25 Hammack pg. 264 14. pg. 268. 1, 2.

• Prove that if $$p$$ is prime then $$\binom{p}{r}$$ is divisible by $$p$$. Use this to prove that $$n^p\equiv n(\mod p)$$ for all $$n$$ and any prime number $$p$$.

• Let $$f(x)=(1+x)^{n}$$. Prove that the $$i^{th}$$ derivative $f^{i}(x)=\frac{n!}{(n-i)!}(1+x)^{n-i}$ and therefore $$f^{i}(0)=n!/(n-i)!$$. Taylor’s formula says that $(1+x)^{n}=\sum_{i=0}^{n}\frac{f^{i}(0)}{i!}x^{i}$ while the binomial theorem says that $(1+x)^{n}=\sum_{i=0}^{n}\binom{n}{i}x^{i}.$ Use these facts to conclude that $$\binom{n}{i}=\frac{n!}{(n-i)!i!}$$

• Oct 28: Gilbert/Vanstone, Chapter 5: 23-32, 24, 29, 30, 32
• Oct 30: GV Chapter 5: 31, 37, 41, 42, 44, 47, 48.

Chapters 12 and 14 of The Book of Proof

• Nov 11: Hammack p. 228: 2, 4, 7, 8; p. 232: 2, 4, 8, 12, 14
• Nov 13: Hammack p. 232: 16, 18. p. 238: 2, 4, 8, 10. p. 241: 2, 6, 8.
• Nov 15: Hammack p. 243: 2, 6, 8, 10.
• Nov 18: Hammack p. 274: 2, 4, 12, 14, 16
• Nov 20: Hammack p. 280: 2, 4, 8, 12, 14
• Nov 22: Hammack p. 280: 10, p. 283: 1, 2, 6, 8, 10.