Section 4.1 Divisibility and Modular Arithmetic 87 CHAPTER 4 Number Theory and Cryptography SECTION 4.1 Divisibility and Modular Arithmetic 2. a) 1 | a since a = 1 · a. b) a | 0 since 0 = a · 0. 4. Suppose a | b ‚ so that b = at for some t ‚ and b | c‚ so that c = bs for some s. Then substituting the first equation into the second‚ we obtain c = (at)s = a(ts). This means that a | c‚ as desired. 6. Under the hypotheses‚ we have c = as and d = bt for some s and t . Multiplying
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------------------------------------------------- Prime number A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime‚ as only 1 and 5 divide it‚ whereas 6 is composite‚ since it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than
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Introduction: Whole Number is very important in the career of Medical Billing and Coding. It is the building blocks of mathematics. Working with whole numbers help you when are adding and subtracting hundreds and thousands of dollars. Then you might have to round up numbers when you are counting money. Multiplying whole numbers are very important‚ it is a repeating of addition. You have to budget Money. Dividing whole number comes when you are multiplying whole numbers II. Body: A. Whole Numbers 1. Place
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n Chinese tradition‚ certain numbers are believed by some to be auspicious (吉利) or inauspicious (不利) based on the Chinese word that the number name sounds similar to. The numbers 0‚ 6‚ 8‚ and 9 are believed to have auspicious meanings because their names sound similar to words that have positive meanings. Contents [hide] 1 Lucky numbers 1.1 Zero 1.2 Two 1.3 Three 1.4 Five 1.5 Six 1.6 Seven 1.6.1 Forty-nine 1.7 Eight 1.8 Nine 2 Unlucky numbers 2.1 Four 2.2 Five 2.3 Six 3 Combinations
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Random number tables have been used in statistics for tasks such as selected random samples. This was much more effective than manually selecting the random samples (with dice‚ cards‚ etc.). Nowadays‚ tables of random numbers have been replaced by computational random number generators. Tables of random numbers have the desired properties no matter how chosen from the table: by row‚ column‚ diagonal or irregularly. The first such table was published by a student of Karl Pearson’s in 1927‚ and
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assessment in 2011 and 2012 STELLAR NUMBERS SL TYPE I Aim: In this task you will consider geometric shapes which lead to special numbers. The simplest example of these are square numbers‚ 1‚ 4‚ 9‚ 16‚ which can be represented by squares of side 1‚ 2‚ 3 and 4. The following diagrams show a triangular pattern of evenly spaced dots. The numbers of dots in each diagram are examples of triangular numbers (1‚3‚6‚)…. 1 3 6 10 15 Complete the triangular numbers sequence with three more terms. Find
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The poem we are analyzing is called "To Autumn" by a poet named John Keats. The poem is an Ode to autumn. It’s a very serious‚ thoughtful poem that praises the season autumn. From the language and words Keats uses‚ we can tell this poem was written some time ago in the early 18th century. The poem is dedicated to autumn and is an expression of joy and harvest. We can tell this poem is an ode because of the way he praises autumn ’Season of mists and mellow fruitfulness.’ The first stanza is mainly
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The Harvard Style is a simple system used internationally by scholars and researchers. This style has two elements: In-text citations In the body of your paper‚ give the surname of the author and the date of publication. Also give the page number if you quote a passage directly or if you paraphrase (put the idea into your own words). List of References At the end of your paper‚ give full publication or internet information so that a reader can easily locate your sources In-text Citations
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literature for children each year. Upon reading “Number the Stars” by Lois Lowry‚ the Newbery Medal winner in 1990‚ one can easily understand why this great novel won the coveted John Newbery Medal. There are many reasons as to why “Number the Stars” obtained the Newbery Medal but two main reasons stick out: The use of italics as a descriptive device‚ and the central theme pertaining to the difficulties of growing up. Firstly‚ Lois Lowry’s “Number the Stars” focuses on Annemarie Johansen‚ a young
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play for his readers‚ the poet incorporates visual images as well as aural effects into his poem. In "Spring and Fall"‚ Hopkins uses rhythm‚ word choice‚ and alliteration to fully integrate the readers‚ as if the readers were right next to Margaret as she undergoes these changes. "Spring and Fall" is a short poem in one stanza; however‚ Hopkins ’ choice of rhythm divides the poem into two distinct sections. The first section illustrates the childlike mind‚ while the second section portrays the grown-up
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