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*If p is prime and p divides a_1a_2...a_n, prove that p divides a_i for some i | *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? | *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 | Problem 25 | ||
Revision as of 19:30, 21 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
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)
