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Use Exercise 37 to show that the integers 235 - 1, 234 -

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 38E Chapter 4.3

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 38E

Problem 38E

Use Exercise 37 to show that the integers 235 - 1, 234 - 1, 233 - 1, 231 - 1, 229 - 1, and 223 - 1 are pairwise relatively prime.

Reference Exercise 37:

        Use Exercise 36 to show that a and b are positive integers, then gcd(2a - 1, 2b - 1) = 2gcd(a,b) - 1. [Hint: Show that the remainders obtained when the Euclidean algorithm is used to compute gcd(2a - 1, 2b - 1) are from 2r - 1, where r  is a remainder arising when the Euclidean algorithm is used to find gcd(a,b).]

Reference Exercise 36:

        Show that if a and b are both positive integers then, (2a - 1) mod (2b - 1) = 2a mod b -1.

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 Week 3 The legislative branch: the congress and the Texas legislature chapter 6 o The concept of representation  The first meaningful use of concept of representation was in the American and French revolutions; it was an important issue when the U.S. Constitution was written  The great compromise resolved this important issue and saved the convention o Representation  The meaning of representation is always under debate. Is it people, geographic area, an ideology, or something else  And should representatives vote as we would vote or as their expertise and values tell them to vote o Roots of the us congress  Section 1.  All legislative Powers herein granted shall be vested in a Congress of the United States, which shall consist of a Senate and House of Representatives.  A term of congress last two years. It begins in an odd ­numbered year and ends in an even­numbered year. Each year is called a session. We are in the second session of 114 congress  Congress has 435 members  Congress is organized by political party. The party that dominates the House provides the leadership for the House, including all committees. The party that dominates the Senate provides the leadership for the Senate, including all committees. o Differences  House Senate  Every house member represents a district  Every senator represents  Serves 2 year a state  House must originate revenue bills­ closer to  Serves 6 year terms people  Approves presidential  House initiates impeachment and passes articles appointments (cabinet, of impeachment courts)  Has a rules committee, which controls debate  Approves treaties power centralized around speaker relatively high  Holds trails on Articles of turnover compared with senate impeachment  435 members based on size of state  Water leadership­ individual senators very powerful  Can filibuster legislation  100 members with two from each state o Leadership  Speaker of the house is elected by the membership, and is always the dominant part. The speaker is next in line for the presidency after the VP  The Vice President is the leader of the senate o Reapportionment and redistributing  First congress was in 1789 and had 65 members  With each census, more members were added until House membership was frozen at 435 in 1910  After each Census, districts have to be redrawn and Members reallocated. The 435 members are reapportioned.  When district lines are redrawn, it is called redistricting  Jerry meandering when districts are drawn politically o The power of incumbency  Members of congress have a good chance of getting elected as many times as they want, they get a seat. Several factors help them:  They service strategy  Project to help the people in the District  Less now due to pork barrel spending  Fundraising  Disadvantages  Angry voters  Scandals  Effects of elections  Turnout issues  Presidential candidates  Superpac spending against the incumbent o Powers of congress  Investigate issues and bring them to national attention (think of homelessness, health care, hunger, decaying infrastructure)  Oversees and investigates the bureaucracy.  Committees can subpoena information and people, cross­examine people, bring criminal charges for contempt and so on  Enumerated taxes  Levy taxes  Borrow money  Regulate commerce  Establish rules for naturalization  Coin money  Punish counterfeiting  Investigated by secret service  Establish post offices  Issue patent and copyrights  Create courts  Declare wars o How time is spent  Attending committee meetings  Working on legislation  Conducting investigations  Carrying out oversight  Working on legislation in other ways  Meeting with lobbyists and representatives of interest in the home district  Talking with other members  Handling cases work  Mostly a staff function  Doing things in home district  Holding town hall meetings  Attending events  Riding in parades  Going to sporting events and school events  Dealing with the media  To stay in the news and in control of message  Working toward reflection  Fundraising  Staying in touch with constituents  Working with party leaders  Enact legislation  A bill is written  A bill is introduced can be started in house or senate  Gets referred to committee  Gets sent to the floor  Can be open or closed  If starts at senate it gets sent to the house rules committee  Set to the house floor  The same version has to be passed in both houses  President signs the bill  Veto (all for nothing)  Pocket veto (congress has to be in session for more than 10 days or else it is vetoed)  Veto override 2/3 majority in house

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Chapter 4.3, Problem 38E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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Use Exercise 37 to show that the integers 235 - 1, 234 -