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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count the number of shirts in your closet. The second way to quantify or measure matter is to determine its mass. For example‚ you can buy bananas by the pound. The third and last way to quantify or measure matter is to determine its volume. For example‚ people by gasoline by the gallon. Chemists use a unit that is a certain number of particles. This unit is known as the mole. A mole is the quantity of anything that has the same number of particles found in 12.000 grams of carbon-12. That number of particles
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Inequalities Maths Compiled By : OP Gupta [+91-9650 350 480 | +91-9718 240 480] Exercise 6.1 Question 1: Solve 24x < 100‚ when (i) x is a natural number (ii) x is an integer Answer The given inequality is 24x < 100. (i) It is evident that 1‚ 2‚ 3‚ and 4 are the only natural numbers less than . Thus‚ when x is a natural number‚ the solutions of the given inequality are 1‚ 2‚ 3‚ and 4. Hence‚ in this case‚ the solution set is {1‚ 2‚ 3‚ 4}. (ii) The integers less than are …–3
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Introduction The number π is a mathematical constant that is the ratio of a circle’s circumference to its diameter‚ and is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century‚ though it is also sometimes written as "pi. π is an irrational number‚ which means that it cannot be expressed exactly as a ratio of any two integers (fractions such as 22/7 are commonly used to approximate π; no fraction can be its exact value); consequently‚ its decimal
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Mobile Number Portability What is MNP? Mobile Number Portability (MNP) allows the Mobile service subscribers to retain their existing mobile number when they move from one Service Provider to another Service Provider with in a same licensed service area. In other words‚ the Mobile Service Subscriber can switch from one operator to other operator with in telecom circle while keeping the same mobile number. MNP has often been considered as a tool to effect increased competition and improve
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concern on the minds of the drivers. Many people die every year due to reckless or careless driving. Driving safe is not a choice it is the law‚ but the law is broken and people suffer the affects of it. Driving drunk and texting while driving is the number one killer while in a vehicle. My perspective before turning sixteen was to drive with out any worries‚ but now because of the incident with my mother I realize driving safe is the right thing to do. She was a strong independent woman since her
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universal precautions. Suppose the spread of a direct contact disease in a stadium is modeled by the exponential equation P(t) = 10‚000/(1 + e3-t) where P(t) is the total number of people infected after t hours. (Use the estimate for e (2.718) or the graphing calculator for e in your calculations.) 1. Estimate the initial number of people infected with the disease. Show how you found your answer. Answer: A total of 474 people would be initially infected. Equation: p(0)=10‚000/(1+3^3) ~ 474
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I chose to analyze Beethoven’s Piano Sonata number 15. Sonata number 15 is opus/work number 28. There are 4 movements in the work‚ but I chose to analyze the first movement. Beethoven was born on December 16‚ 1770 into a family of musicians ("Classical Net - Basic Repertoire List - Beethoven" pars. 1-2). Beethoven began writing music at a young age ("Classical Net - Basic Repertoire List - Beethoven" par. 4). He composed his first original Sonata in 1797 ("Classical Net - Basic Repertoire List -
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Age is just a Number What is age? Does age define who you are? As a nurse in Long term care for fifteen plus years I have learned to look past the number found in the persons chart. I have learned when I as a caregiver look past that number that represents the date the Elder was born; the Elder too tends to look beyond that number and will follow along leading to experiences one might not think Elders of LTC facilities might still accomplish. Here is what Betty has shared with me about her experiences
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pages of double-spaced text (12 point Times New Roman‚ 1-inch margins). Cite the page number(s) for every quotation and use the proper citation style. Since you are discussing only one source here‚ you only need to indicate the page number of the quote in parentheses‚ which should always be placed at the conclusion of the sentence in which the quote appears. For example‚ following the quotation‚ insert the page number in parentheses‚ such as‚ “______________________” (23). You should write the document
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