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| + | Chapter 0: 24, 25, 7, 14, 19, 21 | ||
| + | Due Thursday, January 22 | ||
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| + | -- It's kind of funny that it starts at chapter 0. Very CS of Joe! [[User:eraymond|eraymond]] 13:56, 19 January 2009 (UTC) | ||
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| + | [[Problem 24]] | ||
| + | *If p is prime and p divides a_1a_2...a_n, prove that p divides a_i for some i | ||
| + | *In class we proved this for the case where <math>p|ab</math>. I was unable to extend that proof for <math>n</math> factors of <math>a</math>. Anyone figure this out? | ||
| + | **I updated the page with a [[Problem 24|link]] to the solution. -Nick | ||
| + | |||
| + | Problem 25 | ||
| + | *Use the Generalized Euclid's Lemma to establish the uniqueness portion of the Fundamental Theorem of Arithmetic | ||
| + | |||
| + | Problem 7 | ||
| + | *Show that if a and b are positive integers, then ab = lcm(a, b) * gcd (a,b) | ||
| + | *I completed this problem by writing a and b as prime factorizations, with the gcd and lcm having the min and max of their exponents respectively. --[[User:Podarcze|Podarcze]] 15:15, 20 January 2009 (UTC) | ||
| + | |||
| + | Problem 14 | ||
| + | *Show that 5n + 3 and 7n + 4 are relatively prime for all n | ||
| + | *You can use Euclid's Algorithm to show that <math>gcd(5n+3,7n+4)=1</math>. Start with <math>(7n+4)=1*(5n+3)+(2n+1)</math> and iterate until the remainder is 0. | ||
| + | |||
| + | Problem 19 | ||
| + | *Prove that there are infinitely many primes. (hint: use ex. 18) | ||
| + | |||
| + | [[Problem 21]] | ||
| + | *For every positive integer n, prove that a set with exactly n elements has exactly 2^n subsets (counting the empty set and the entire set) | ||
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| + | --[[User:Aifrank|Aifrank]] 13:56, 18 January 2009 (UTC) | ||
Revision as of 13:54, 22 January 2009
Chapter 0: 24, 25, 7, 14, 19, 21
Due Thursday, January 22
-- It's kind of funny that it starts at chapter 0. Very CS of Joe! eraymond 13:56, 19 January 2009 (UTC)
- If p is prime and p divides a_1a_2...a_n, prove that p divides a_i for some i
- In class we proved this for the case where $ p|ab $. I was unable to extend that proof for $ n $ factors of $ a $. Anyone figure this out?
- I updated the page with a link to the solution. -Nick
Problem 25
- Use the Generalized Euclid's Lemma to establish the uniqueness portion of the Fundamental Theorem of Arithmetic
Problem 7
- Show that if a and b are positive integers, then ab = lcm(a, b) * gcd (a,b)
- I completed this problem by writing a and b as prime factorizations, with the gcd and lcm having the min and max of their exponents respectively. --Podarcze 15:15, 20 January 2009 (UTC)
Problem 14
- Show that 5n + 3 and 7n + 4 are relatively prime for all n
- You can use Euclid's Algorithm to show that $ gcd(5n+3,7n+4)=1 $. Start with $ (7n+4)=1*(5n+3)+(2n+1) $ and iterate until the remainder is 0.
Problem 19
- Prove that there are infinitely many primes. (hint: use ex. 18)
- For every positive integer n, prove that a set with exactly n elements has exactly 2^n subsets (counting the empty set and the entire set)
--Aifrank 13:56, 18 January 2009 (UTC)
