By Kraft, James S.; Washington, Lawrence C

ISBN-10: 1482214423

ISBN-13: 9781482214420

ISBN-10: 1931962022

ISBN-13: 9781931962025

IntroductionDiophantine EquationsModular ArithmeticPrimes and the Distribution of PrimesCryptographyDivisibilityDivisibilityEuclid's Theorem Euclid's unique evidence The Sieve of Eratosthenes The department set of rules the best universal Divisor The Euclidean set of rules different BasesLinear Diophantine EquationsThe Postage Stamp challenge Fermat and Mersenne Numbers bankruptcy Highlights difficulties distinct FactorizationPreliminaryRead more...

summary: IntroductionDiophantine EquationsModular ArithmeticPrimes and the Distribution of PrimesCryptographyDivisibilityDivisibilityEuclid's Theorem Euclid's unique facts The Sieve of Eratosthenes The department set of rules the best universal Divisor The Euclidean set of rules different BasesLinear Diophantine EquationsThe Postage Stamp challenge Fermat and Mersenne Numbers bankruptcy Highlights difficulties precise FactorizationPreliminary effects the elemental Theorem of mathematics Euclid and the elemental Theorem of ArithmeticChapter Highlights difficulties purposes of distinctive Factorization A Puzzle Irrationality

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**Example text**

Convert the base 10 number 1671 to base 2. 1671 835 417 208 104 52 26 13 6 3 1 = = = = = = = = = = = 835 · 2 417 · 2 208 · 2 104 · 2 52 · 2 26 · 2 13 · 2 6·2 3·2 1·2 0·2 + + + + + + + + + + + 1 1 1 0 0 0 0 1 0 1 1. 32 Chapter 1 Divisibility So, 167110 = 110100001112 . Example. It’s always a good idea to make sure that any mathematical method works for an example where you already know the answer. ” So, let’s take a base 10 number, say 314159, and use the above algorithm to “convert” it to base 10: 314159 31415 3141 314 31 3 = = = = = = 31415 · 10 3141 · 10 314 · 10 31 · 10 3 · 10 0 · 10 + + + + + + 9 5 1 4 1 3.

Fermat primes occur in compass and straightedge constructions in geometry. Using only a compass and a straightedge, it is easy to make an equilateral triangle or a square. It’s a little harder to make a regular pentagon, but it’s possible. The constructions of equilateral triangles and regular pentagons can be combined to produce a regular 15-gon. Moreover, by bisecting angles, it’s easy to double the number of sides of a polygon that is already constructed, so 30-sided, 60-sided, and 120-sided regular polygons can be constructed.

Let’s do another example with 23 people voting. Let’s say the chair’s random number is 27938. Now, committee members add 24 if they vote Yes and 1 if they vote No. We’ll tell you what the votes were so that you can see why the method works. Let’s say there are 16 Yes votes, 5 No votes, and 2 abstentions. Then the chair receives the number 27938 + 16 · 24 + 5 + 2 · 0 = 27938 + 389 = 28327. 20 Chapter 1 Divisibility Of course, when the chair subtracts 27938 from 28327 the answer is 389, and the Division Algorithm says that 389 = 16 · 24 + 5.

### An Introduction to Number Theory with Cryptography by Kraft, James S.; Washington, Lawrence C

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