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[[Category:MA453Spring2009Walther]] | [[Category:MA453Spring2009Walther]] | ||
| + | [[Week 2]]-Jan 22 (In[tro]duction, 0): Ch 0: 25, 24, 7, 14, 19, 21 | ||
Chapter 0: 24, 25, 7, 14, 19, 21 | Chapter 0: 24, 25, 7, 14, 19, 21 | ||
Due Thursday, January 22 | Due Thursday, January 22 | ||
Revision as of 14:55, 22 January 2009
Week 2-Jan 22 (In[tro]duction, 0): Ch 0: 25, 24, 7, 14, 19, 21
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)
