Sonography Career Free Essays

Sonography The first time I started college I had a different career path. It was not after my pregnancies that I decided to make the change to sonography. I think sonography is a career that really fits my expectations, and to make sure I am in the right path I will research this career in depth. We will write a custom essay sample on Sonography Career or any similar topic only for you Order Now Tests and evaluations in sonography are in writing and practical. They are prepared to measure the student’s understanding of the material, and the knowledge in manipulating the transducer (Craig 53). At the beginning I was curious about how the test would be. I now know the reading tests are normal types of tests: objective, which is cover multiple choices or true and false, and the subjective, which are the essays questions. I also found, I will have some tests on computer (Craig 46-47) which I think is a great idea because I will get accustomed to the way that certifying test are now presented, also it will help me to be successful in my career. The practical exams will help me to familiarize in working with patients on a day to day work habit, and the writing and computer exams will prepare me for my certifying examination post graduation. Sonography has different types of sonograms in which I can specialize. There are eight types of diagnostic ultrasounds specialties, but there are three which are the common practiced: abdominal, obstetric and gynecologic, and cardiac (Craig 152). This is great for me, because obstetric and gynecologic is the type of ultrasound I would like to specialize. Also, since this type of ultrasound is one of the most common I think it will be easier find a job in this field. Registration is very important in the sonography field. Graduates are encouraged to take their registry examinations as soon as possible after completion of their training (Craig 53). † In my opinion this is great because the knowledge of the sonographer is tested again giving us the opportunities to have prepared sonographers in the health facilities. Since â€Å"employers prefer candidates registered through the American Registry for Diagnostic Medical Sonography (Morsch)† being registry is an open door for sonographers trying to get a job. American Registry for Diagnostic Medical Sonography (ARDMS) offers the opportunity to sonographers to earn three different types of credentials. The credentials are Registered Diagnostic Medical Sonographer, Registered Diagnostic Cardiac Sonographer, and Registered Vascular Technologist. In order to be award with one of them the graduate needs to pass two exams: one general and one specialty exam that depend on the credential wanted (ARDMS). This I think is a wonderful requisite because it elevates the prestige of sonographers. As more credentials I can get more knowledge I will have. Salary is another point in my research. â€Å"In 2008, the starting salary for a diagnostic medical sonographer was $43,600 (Diagnostic Medical Sonography). Even though this is not the most important point for me I have to admit it is within the range I was expecting for salary. According to the Bureau of Labor Statistics the annual mean wage in Florida is $56,990 which breaks down in $27. 40 an hour (Bureau). This is better than the figures from 2008 and if salary follows this line by the time I graduate it should be even greater. The outlook of this job is great. â€Å"Employment of diagnostic medical sonographers is expected to increase by about 18 percent through 2018-faster than average for all occupations (Bureau)†. This is about 2. 25 percent per year. Sonograms will become the most popular tool in the medicine field to check the human body since is not harmful and very effectively. The truth is that â€Å"sonography is in high demand trough out the country since the uses of sonography are expanded (Sonography). † Another factor that will help on the growth of the sonography is the population who is aging. Older people needs more tests and Florida is one of the favorite states for retired people, so I would say that we will have a nice increase on the growth rate. Learn about searching jobs I think is important. After I graduate I would like to know the most important factors when looking for a job. As I found there are many tips for a successful job search. A resume accompanied of a personalized letter, always listen to the interviewer and answer the questions straight and precise, choose professional clothes for the interview day, negotiate the salary if it is no right for you, and always get the final offer in writing (Craig 213-216). Responsibilities are part of all jobs. Sonographers once they graduate start taking responsibilities which include preparing reports, organized daily work schedule, and report equipment failure(Craig 205). Searching this I notice that sonographers don’t only make reports and scans in fact I will do more than what I thought was my responsibilities. Sonography also has disadvantages. Musculoskeletal injury has been affecting sonographers for years. The amount of sonographers in the country is a small number compare to other professions in health, but the amount of injured sonographers is that big relative to the total number of sonographers that brought the attention of NIOSH. To protect sonographers from musculoskeletal pain NIOSH recommend a few safety measures for under graduates to avoid injuries (Musculoskeletal). This is really worry because I want a profession not an illness. I really think the programs should emphasize more on safety rules to avoid MSI from happen so frequently. I have learned plenty of information with my sonography research paper. Now I like even more my career. I believe I will have a great future if I keep going in this direction. Sonography is a very popular job, and learn the inside of the human body it is always amazing. [1010] Work Cited ARDMS,†Credential and Examinations. †Web. 15 Nov. 2011. https://ardms. org/credentials_examinations/ Bureau of Labor Statistics, U. S. Department of Labor, Occupational Outlook Handbook,2010-11edition, Diagnostic Medical Sonographers. Web. 9 Nov. 2011. http://www. ls. gov/oco/ocos273. htm Craig Marveen,†Essentials of Sonography and Patient Care†. St. Louis: Saunders,2006. Print. â€Å"Diagnostic Medical Sonography. † CAAHEP. Org. CAAHEP, 2009. Web. 3 Oct. 2011. †¹ http://www. caahep. org/content. aspx? ID=23 Morsch,Laura. †10 Best Paying Jobs for Associate Degress. †CNN. com. Cable News Network,05 May 2006. Web. 3 Oct. 2011. http://www. cnn. com/2006/us /careers/05/04/cb. associates â€Å"Musculoskeletal pain hits 80% of sonographers. † Occupational Health Management 1 Feb. 2007. Academic OneFile. Web. 4 Oct. 2011. How to cite Sonography Career, Papers

Operation Management in Gas Pipeline Construction in Australia

Question: Discuss about the Operation Management in Gas Pipeline Construction in Australia. Answer: Research design Research question For diminishing the issues of operation management in the gas pipeline industry of Australia, it is necessary for the research designer to consider all the construction aspects accurately. The issues could be resolved if after the completion of the research study the designer become able to answer all the questions properly. The research questions should consider both the internal ad the external environment. The developed question or the research area is as follows: What are the different operation management strategies should develop in gas pipeline industry of Australia? What is the reason behind the application of pipe leakage detection program in the gas pipeline industry of Australia? Identify different API leak detection programs those are essential for the development of pipeline guidance programs? Related work Different types of experimental design approaches are there such as experimental, quasi, co-relational, ethnography etc. With the help of these research design approaches the researcher would be able to evaluate the project successfully. After analyzing the outline of the research topic that is Role of operational management in pipeline industries of Australia, it is important to select the most suitable research design approaches correlational and quasi research design. However, based on the positive and negative impact of these research design approaches, two of these approaches are selected by the researcher. Advantages and disadvantages of quasi research design Advantages disadvantages This particular approach is useful when it is unethical to influence the Independent Variable. The rate of human level error could be minimized with the application of this particular research design approach. It helps to study the real effect so that the realism and ecological validity can be increased accordingly. In certain occasions the irrelevant variables could create artificial situation. These approaches have less amount of internal validity. The participants Are required to wait until the independent variables are identified by the researchers. The control over the variables will also increase after the implementation of the experimental research work. In the secondary data collection process the participants who are taking part in the research approach, are influenced with the environment. The relationship between the cause and effect in a much easier way could be determined. Another disadvantage of this research design implies that, manipulation of the variables is not seen as completely objectives in nature. A much better level of experimental resultant can be generated with the help of this research design. This particular approach could be used only in the naturally variable conditions. Advantages and disadvantages of Correlational research design Advantages Disadvantages The correlation research design approach allows the researchers to gather information than the experiments. On the other hand it basically take place outside the laboratory thus, the generated results are referred to as much applicable in the daily usage. This approach is unable to provide conclusion statement or conclusive reasons rather it is only helps to uncover the relationship between the researchers and the participants in the secondary data collection research. For further research development it is also very much helpful. This research design is unable to expose those variables which are influencing in nature and unable to identify those variables which influence the others. It helps to provide a more valuable and reasonable situation to the researchers for investigation and research work. The causations could not be determined with the help of this particular approach. Design Different research design approaches are there and after considering the research topic it has been defined that, for this particular topic, quasi experiment and correlation research design are very much beneficial. In order to answer all of these Questions it is necessary to use the correlation and quasi research design approaches. In order to develop the design the positivism research philosophy should be developed by the researchers and on the other hand with the help of this particular design approach the level of understanding on this particular topic could be extended and the hidden facts about proper decision making approach will be completely done with the help of this approach. Though while collecting data from the resources it is necessary for the research developers to consider the primary data collection. That is development of survey and set of questionnaires. Te answers gained from the participants will help the researchers to design the set of policies and procedures of operational management for the gas pipelines Industries of Australia. For this particular construction plan the researcher should consider, the chemical nature and quality of the gas that is going to pass through the pipelines. It will help the project constructors to understand about the specific type of materials and materials th ose might be used for the development of the pipelines of Australia. On the other hand it also helps the researcher to develop proper decision making approaches regarding the development purposes. In order to develop the design for this accurate research topic that is operation management in Pipeline construction in Australia the identified strength and limitation of the research design are as follows: It has been defined that all the hidden facts are not accurately revealed those are needed for the development of operational management of gas pipelines construction in Australia. However, it took lower amount of cost and time as well for the development. After considering all of these approaches it has been defined that in order to develop answers for this particular research question quasi and correlation research design approach is required to be developed. Bibliography Best, J.W. and Kahn, J.V., 2016.Research in education. Pearson Education India. Campbell, D.T. and Stanley, J.C., 2015.Experimental and quasi-experimental designs for research. Ravenio Books. Donofrio, B.M., Lahey, B.B., Turkheimer, E. and Lichtenstein, P., 2013. Critical need for family-based, quasi-experimental designs in integrating genetic and social science research.American journal of public health,103(S1), pp.S46-S55. Gast, D.L. and Ledford, J.R. eds., 2014.Single case research methodology: Applications in special education and behavioral sciences. Routledge. Goodwin, C.J., 2016.Research in psychology methods and design. John Wiley Sons. Mackey, A. and Gass, S.M., 2015.Second language research: Methodology and design. Routledge. Mertens, D.M., 2014.Research and evaluation in education and psychology: Integrating diversity with quantitative, qualitative, and mixed methods. Sage publications. Moring, B., 2014.Research methods in psychology: Evaluating a world of information. WW Norton Company.

Adventures Essays (467 words) - Rafting, , Term Papers

Adventures On The Rapids "This could be your last meal," my mother jokingly said before we left that day. The day was bright, and the sun gleaming. The group packed into the muggy van, it was stifling hot, and downright uncomfortable. On a hot summer Missouri day, in the middle of July, your shirt clung to your damp skin. The mission: White Water Rafting. The drive to the river, where we would begin our incredible journey, seemed to last endless miles. The humidity was almost unbearable; the van had absolutely no ventilation. Fifteen people packed like sardines, anticipating beginning the trip over the rapids. Envisioning the cool water splashing on our sunburned faces. Fleeing from the van was like jumping from a burning building. Bathing suits on, sunblock applied, strength in tact, we were set. Waiting for the moment when we were able to jump into the raft, and head down stream. Savoring our feet splashing in the chilly water. What we didn't know, was what the day was about to become, and how it could have changed our lives forever. Less than two hours from now, we would know. I was assigned to a raft with my brother, my friend, and the river guide. The adults went in another. About an hour after we left, we made our first stop; an enormous rock midstream. We sat there for several minutes hopped back into the raft and we were on our way, rushing down the river, nearing towards the end. As we approached the last of the rapids, our guide asked if we wanted to surf up them. Surfing is basically paddling up a rapid. We practiced for several minutes, because you have to get used to paddling against the current, then you can go up against a rapid. We were prepared, and ready for the last of the excitement in our adventure on the rapids. Fighting our way up the rapid, it all played out in slow motion. We paddled hard, and so powerfully. I remember it being like a space ship bursting into the sky, or a bomb exploding, when another raft came charging down the rapid, crashing into my side of the raft. I was thrown off the raft, into the bitterly frigid water. All playing back in my head now, even slower I found myself panicking. I realized that my foot had been lodged in between two rocks. As my life is flashing before my eyes, I saw fearful people above the water. I was unbelievably frightened. I heard screams, and yelling from above water. Then I thought to myself, What if I can't see images of my life seconds from now? What if I can't feel terrified? What if I can't see the people just above me? Abruptly, as if I was released from a funnel cloud. I was pushed into the stream. Now I was gliding downstream. My foot was released, my fear calmed, and the images weren't flashing anymore. I was safe.

Free Essays on Interview With The Amish

Interview With James Shoemacher Amish Vs. American James Bernard Shoemacher, Æ’Â ±JBÆ’Â °, has been a friend and business associate of mine for about three years. He was born and raised in an Amish community in New Harmony, Indiana. JB spent twenty three years inside the community until he was married, then he left to join his brother who lived closer to town than the community was. Together they make up the New Harmony carriage MFR., this is how I came into contact with him. I first met JB and his brother Jed in a draft horse auction in Waverley In., he was there with several of their hand crafted carriages. I got to talking with him and soon thereafter started a business relationship with him that has lasted already five years. Over the time that IÆ’Â ­ve know JB I took in a lot of knowledge about his culture and heritage, thus making him a prime suspect for my interview. That, and IÆ’Â ­ve always been intrigued by the Amish people. After talking with JB for a while I started to understand his English diale ct as if it were the country bumpkin slang I have grown to love. Premeditating the oncoming interview I developed a series of twelve questions that I hoped would help me better understand the Amish people and there never changing way of life. I wanted to be able to sympathize with the man in the horse-drawn buggie on the side of the road and be able to see the world from their perspective. As we sat in one of the finished carriages in JBÆ’Â ­s barn I began my quest for enlightenment. First I asked, what are the main differences between the Amish and American cultures? He stated that the Amish teach separatism and that they are not allowed to go to war, take oaths, or hold any public offices. They are not allowed to use any kind of electrical element inside the home, E.G. telephones, computers, and televisions. Amish people focus a lot on togetherness, the whole community will be involved in what has been made po... Free Essays on Interview With The Amish Free Essays on Interview With The Amish Interview With James Shoemacher Amish Vs. American James Bernard Shoemacher, Æ’Â ±JBÆ’Â °, has been a friend and business associate of mine for about three years. He was born and raised in an Amish community in New Harmony, Indiana. JB spent twenty three years inside the community until he was married, then he left to join his brother who lived closer to town than the community was. Together they make up the New Harmony carriage MFR., this is how I came into contact with him. I first met JB and his brother Jed in a draft horse auction in Waverley In., he was there with several of their hand crafted carriages. I got to talking with him and soon thereafter started a business relationship with him that has lasted already five years. Over the time that IÆ’Â ­ve know JB I took in a lot of knowledge about his culture and heritage, thus making him a prime suspect for my interview. That, and IÆ’Â ­ve always been intrigued by the Amish people. After talking with JB for a while I started to understand his English diale ct as if it were the country bumpkin slang I have grown to love. Premeditating the oncoming interview I developed a series of twelve questions that I hoped would help me better understand the Amish people and there never changing way of life. I wanted to be able to sympathize with the man in the horse-drawn buggie on the side of the road and be able to see the world from their perspective. As we sat in one of the finished carriages in JBÆ’Â ­s barn I began my quest for enlightenment. First I asked, what are the main differences between the Amish and American cultures? He stated that the Amish teach separatism and that they are not allowed to go to war, take oaths, or hold any public offices. They are not allowed to use any kind of electrical element inside the home, E.G. telephones, computers, and televisions. Amish people focus a lot on togetherness, the whole community will be involved in what has been made po...

Tesco in Indonesia Essay Example | Topics and Well Written Essays - 1500 words

Tesco in Indonesia - Essay Example 1). Due to their nature of co-operation and the willingness to share, competition is not very common, as it is viewed as a practice that benefits oneself. Nevertheless, business is evident in Indonesia with competitive industries from within and foreign; however, the government of Indonesia is very keen on how businesses operate such that, laws that ensure fair competition have been put in place. For instance, Law number 5 prohibits monopolistic practices and unfair business competition. The objectives of Indonesia competitive law include safeguarding the interests of the public by improving the economic efficiency. Secondly, it is geared towards creating a conducive business climate by ensuring fair business competition. Thirdly, the law prevents monopolistic practices, which may lead to unfair business practices, and finally, it ensures effectiveness and efficiency in business operations. In addition, Indonesia gives room from foreign businesses to operate in their country, as long as they abide to the rules and regulations governing business practices (Act number 5, p1). Historical background of Tesco Tesco started its operations in 1919, when its owner Jack Cohen started selling groceries in a stall, which was located in London, with only a ?1 on the first day. In 1924, Tesco become a brand after acquiring tea shipment and in 1930, Mr. Cohen built a warehouse in North London declaring Tesco a private company. Over the years, it bought rival shops, and by the 60s, the supermarket was already selling a variety of products. Tesco entered Guinness book of records as the largest store in Europe, and in the 70’s, Tesco had opened a petrol station, which was an independent retailer. In the 90’s, Tesco had already completed several takeovers, while in 1996, it introduced a 24-hour store (The telegraph Para. 10). Today, Tesco has grown to a giant supermarket, and it is visible globally. Its diversification is evident in the United States, Indonesia, La tin America, china, Scotland, china, Norway, Middle East and Mexico. In addition, Tesco is committed to corporate social responsibility such as charities for schools and communities. The company has incorporated technology and it encourages online shopping. In addition, the supermarket offers a range of products to meet the customer’s needs. Business culture and social preferences of Indonesia To operate successfully in Indonesia, a corporate requires understanding the business culture and laws governing businesses this country. Therefore, establishing of relationships is necessary; however, it takes time. Loyalty and trustworthiness are a necessity in maintaining a relationship, especially with the government officials. In this country, men are considered mature if they are over the age of 40 years; therefore, a foreign country may consider sending older executives rather than the young ones, especially in negotiations (Doing Business in Indonesia, p112). In addition, busine sses in Indonesia are required to develop their surrounding community; therefore, corporate social responsibility is encouraged. Firms are encouraged to recruit carefully for their positions, as unsatisfactory performance is not regarded enough reason to lay off an employee. Dismissal for misconduct is effective only after three warnings. Developing strong relationships with

Carrer management Essay Example | Topics and Well Written Essays - 1500 words

Carrer management - Essay Example With my psychometric result, I have further explored that ESFJs also have the quality to take good care of things that are in their custody (Butt 2009). They are excellent protectors and can gain remarkable achievements in education and medical field. Functional analysis of ESFJs shows that they have extraverted feelings and they are talented enough to adapt to new environments quickly, most specifically they have command over e-world gadgets and systems (Cazenovia 2012). In addition, they can easily express a variety of feelings in front of others and mostly they are motivated or de-motivated with others response towards their thoughts and actions. Further, it has been noticed that ESFJ are usually career oriented people and they do have passion to do something remarkable in the aspect of academic or professional ground. Additionally, they seek interesting as well as satisfying careers and to do this they work with deep concentration and intend to attempt all processes in a thoughtf ul manner. Competitive partner or environment works as a catalyst for them because they always possess passion to defeat all their enemies and competitors and in order to achieve this purpose they can do anything, which is possible and accessible. Personal swot analysis illustrates my strengths, followed by my weaknesses, then opportunities, which will assist me during my journey towards goals and the threats that might be a barrier in the path towards my objective. My biggest strength is that I am a business management undergraduate student at PSB institute and I am only a few steps away from my completing my degree. I know it is my greatest strength and it will help me in the near future to take initiative in the business world (Psni 2012). I am very passionate about my career and I am confident that my talent, skills, and willingness to achieve my

Eulers Totient Theorem

Eulers Totient Theorem Summary   Ã‚   Euler Totient theorem is a generalized form of Fermats Little theory. As such, it solely depends on Fermats Little Theorem as indicated in Eulers study in 1763 and, later in 1883, the theorem was named after him by J. J. Sylvester. According to Sylvester, the theorem is basically about the alteration in similarity. The term Totient was derived from Quotient, hence, the function deals with division, but in a unique way. In this manner, The Eulers Totient function à Ã¢â‚¬   for any integer (n) can be demarcated, as the figure of positive integers is not greater than and co-prime to n. aà Ã¢â‚¬  (n) = 1 (mod n) Based on Leonhard Eulers contributions toward the development of this theorem, the theory was named after him despite the fact that it was a generalization of Fermats Little Theory in which n is identified to be prime. Based on this fact, some scholarly source refers to this theorem as the Fermats-Euler theorem of Eulers generalization. Introduction I first developed an interest in Euler when I was completing a listener crossword; the concealed message read Euler was the master of the crossword. When I first saw the inclusion of the name Euler on the list of prompt words, I had no option but to just go for it. Euler was a famous mathematician in the eighteenth century, who was acknowledged for his contribution in the mathematics discipline, as he was responsible for proving numerous problems and conjectures. Taking the number theory as an example, Euler successively played a vital role in proving the two-square theorem as well as the Fermats little theorem (Griffiths and Peter 81). His contribution also paved the way to proving the four-square theorem. Therefore, in this course project, I am going to focus on his theory, which is not known to many; it is a generalization of Fermats little theorem that is commonly known as Eulers theorem. Theorem Eulers Totient theorem holds that if a and n are coprime positive integers, then since ÃŽÂ ¦n is a Eulers Totient function. Eulers Totient Function Eulers Totient Function (ÃŽÂ ¦n) is the count of positive integers that are less that n and relatively prime to n. For instance, ÃŽÂ ¦10 is 4, since there are four integers, which are less than 10 and are relatively prime to 10: 1, 3, 7, 9. Consequently, ÃŽÂ ¦11 is 10, since there 11 prime numbers which are less than 10 and are relatively prime to 10. The same way, ÃŽÂ ¦6 is 2 as 1 and 5 are relatively prime to 6, but 2, 3, and 4 are not. The following table represents the totients of numbers up to twenty. N ÃŽÂ ¦n 2 1 3 2 4 2 5 4 6 2 7 6 8 4 9 6 10 4 11 10 12 4 13 12 14 6 15 8 16 8 17 16 18 6 19 18 20 8 Some of these examples seek to prove Eulers totient theorem. Let n = 10 and a = 3. In this case, 10 and 3 are co-prime i.e. relatively prime. Using the provided table, it is clear that ÃŽÂ ¦10 = 4. Then this relation can also be represented as follows: 34 = 81 à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mode 10). Conversely, if n = 15 and a = 2, it is clear that 28 = 256 à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod 15). Fermats Little Theory According to Liskov (221), Eulers Totient theorem is a simplification of Fermats little theorem and works with every n that are relatively prime to a. Fermats little theorem only works for a and p that are relatively prime. a p-1 à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod p) or a p à ¢Ã¢â‚¬ °Ã‚ ¡ a (mod p) where p itself is prime. It is, therefore, clear that this equation fits in the Eulers Totient theorem for every prime p, as indicated in ÃŽÂ ¦p, where p is a prime and is given by p-1. Therefore, to prove Eulers theorem, it is vital to first prove Fermats little theorem. Proof to Fermats Little Theorem As earlier indicated, the Fermats little theorem can be expressed as follows: ap à ¢Ã¢â‚¬ °Ã‚ ¡ a (mod p) taking two numbers: a and p, that are relatively prime, where p is also prime. The set of a {a, 2a, 3a, 4a, 5aà ¢Ã¢â€š ¬Ã‚ ¦(p-1)a}à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦(i) Consider another set of number {1, 2, 3, 4, 5à ¢Ã¢â€š ¬Ã‚ ¦.(p-1a)}à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦(ii) If one decides to take the modulus for p, each element of the set (i) will be congruent to an item in the second set (ii). Therefore, there will be one on one correspondence between the two sets. This can be proven as lemma 1. Consequently, if one decides to take the product of the first set, that is {a x 2a x 3a x 4a x 5a x à ¢Ã¢â€š ¬Ã‚ ¦. (p-1)a } as well as the product of the second set as {1 x 2 x 3 x 4 x 5à ¢Ã¢â€š ¬Ã‚ ¦ (p-1)}. It is clear that both of these sets are congruent to one another; that is, each element in the first set matches another element in the second set (Liskov 221). Therefore, the two sets can be represented as follows: {a x 2a x 3a x 4a x 5a x à ¢Ã¢â€š ¬Ã‚ ¦. (p-1)a } à ¢Ã¢â‚¬ °Ã‚ ¡ {1 x 2 x 3 x 4 x 5à ¢Ã¢â€š ¬Ã‚ ¦ (p-1)} (mode p). If one takes out the factor a p-1 from the left-hand side (L.H.S), the resultant equation will be Ap-1 {a x 2a x 3a x 4a x 5a x à ¢Ã¢â€š ¬Ã‚ ¦. (p-1)a } à ¢Ã¢â‚¬ °Ã‚ ¡ {1 x 2 x 3 x 4 x 5à ¢Ã¢â€š ¬Ã‚ ¦ (p-1)} (mode p). If the same equation is divided by {1 x 2 x 3 x 4 x 5à ¢Ã¢â€š ¬Ã‚ ¦ (p-1)} when p is prime, one will obtain a p à ¢Ã¢â‚¬ °Ã‚ ¡ a (mod p) or a p-1 à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod p) It should be clear that each element in the first set should correspond to another element in the second set (elements of the set are congruent). Even though this is not obvious at the first step, it can still be proved through three logical steps as follows: Each element in the first set should be congruent to one element in the second set; this implies that none of the elements will be congruent to 0, as pand a are relatively prime. No two numbers in the first set can be labeled as ba or ca. If this is done, some elements in the first set can be the same as those in the second set. This would imply that two numbers are congruent to each other i.e. ba à ¢Ã¢â‚¬ °Ã‚ ¡ ca (mod p), which would mean that b à ¢Ã¢â‚¬ °Ã‚ ¡ c (mod p) which is not true mathematically, since both the element are divergent and less than p. An element in the first set can not be congruent to two numbers in the second set, since a number can only be congruent to numbers that differ by multiple of p. Through these three rules, one can prove Fermats Little Theorem. Proof of Eulers Totient Theorem Since the Fermats little theorem is a special form of Eulers Totient theorem, it follows that the two proofs provided earlier in this exploration are similar, but slight adjustments need to be made to Fermats little theorem to justify Eulers Totient theorem (KrÃÅ'Å’iÃÅ' zÃÅ'Å’ek 97). This can be done by using the equation a ÃŽÂ ¦n à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod n) where the two numbers, a and n, are relatively prime, with the set of figures N, which are relatively prime to n {1, n1. n2à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n }. This set is likely to have ÃŽÂ ¦n element, which is defined by the number of the relatively prime number to n. In the same way, in the second set aN, each and every element is a product of a as well as an element of N {a, an1, an2, an3à ¢Ã¢â€š ¬Ã‚ ¦anÃŽÂ ¦n}. Each element of the set aN must be congruent to another element in the set N (mode n) as noted by the earlier three rules. Therefore, each element of the two sets will be congruent to each other (Giblin 111). In this scenario case, it can be said that: {a x an1 x an2 x an3 x à ¢Ã¢â€š ¬Ã‚ ¦. an ÃŽÂ ¦n } à ¢Ã¢â‚¬ °Ã‚ ¡ {a x   n1 x n2 x n3 x à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n } (mod n). By factoring out a and aÃŽÂ ¦n from the left-hand side, one can obtain the following equation a ÃŽÂ ¦n {1 x n1 x n2 x n3 x à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n} à ¢Ã¢â‚¬ °Ã‚ ¡ {1 x n1 x n2 x n3 x à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n } (mod n) If this obtained equation is divided by {1 x n1 x n2 x n3 x à ¢Ã¢â€š ¬Ã‚ ¦.n ÃŽÂ ¦n } from both sides, all the elements in the two sets will be relatively prime. The obtained equation will be as follows: a ÃŽÂ ¦n à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod n) Application of the Eulers Theorem Unlike other Eulers works in the number theory like the proof for the two-square theorem and the four-square theorem, the Eulers totient theorem has real applications across the globe. The Eulers totient theorem and Fermats little theorem are commonly used in decryption and encryption of data, especially in the RSA encryption systems, which protection resolves around big prime numbers (Wardlaw 97). Conclusion In summary, this theorem may not be Eulers most well-designed piece of mathematics; my favorite theorem is the two-square theorem by infinite descent. Despite this, the theorem seems to be a crucial and important piece of work, especially for that time. The number theory is still regarded as the most useful theory in mathematics nowadays. Through this proof, I have had the opportunity to connect some of the work I have earlier done in discrete mathematics as well as sets relation and group options. Indeed, these two options seem to be among the purest sections of mathematics that I have ever studied in mathematics. However, this exploration has enabled me to explore the relationship between Eulers totient theorem and Fermats little theorem. I have also applied knowledge from one discipline to the other which has broadened my view of mathematics. Works Cited Giblin, P J. Primes, and Programming: An Introduction to Number Theory with Computing. Cambridge UP, 1993. Print. Griffiths, H B, and Peter J. Hilton. A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation. London: Van Nostrand Reinhold Co, 1970. Print. KrÃÅ'Å’iÃÅ' zÃÅ'Å’ek, M., et al. 17 Lectures on Fermat Numbers: From Number Theory to Geometry. Springer, 2001. Print. Liskov, Moses. Fermats Little Theorem. Encyclopedia of Cryptography and Security, pp. 221-221. Wardlaw, William P. Eulers Theorem for Polynomials. Ft. Belvoir: Defense Technical Information Center, 1990. Print.