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 precise 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 exact FactorizationPreliminary effects the elemental Theorem of mathematics Euclid and the basic Theorem of ArithmeticChapter Highlights difficulties purposes of special Factorization A Puzzle Irrationality

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**Sample text**

If gcd(a, b) = 1, then there exist integers x, y with ax + by = 1. 4. Finding solutions of linear diophantine equations ax + by = c using the Extended Euclidean Algorithm. 1: Divisibility 1. Show that 5 | 120, 2. Show that 7 | 7, 11 | 165, 10 25, and 14 | 98. and 32 | −160. 3. Find all positive divisors of the following integers: (a) 20 (b) 52 (c) 195 (d) 203 4. Find all positive divisors of the following integers: (a) 12 (b) 13 (c) 15 (c) 16 5. Prove or give a counterexample for the following statements: (a) If c | a and c | b, then c | ab.

8. Find an n with 1 < n < 10 such that gcd(n, 60) = 1. 9. Let n be an integer. Show that gcd(n, n + 3) = 1 or 3, and show that both possibilities occur. 7 The Euclidean Algorithm The Euclidean Algorithm is one of the oldest and most useful algorithms in all of number theory. It is found as Proposition 2 in Book VII of Euclid’s Elements. One of its features is that it allows us to compute gcd’s without factoring. In cryptographic situations, where the numbers often have several hundred digits and are hard to factor, this is very important.

In our example, the multiples of the primes 2, 3, 5, 7 created a net. The numbers that fell through this net are the prime numbers. CHECK YOUR UNDERSTANDING 4. Use the Sieve of Eratosthenes to compute the prime numbers less than 20. 5 The Division Algorithm If a and b are integers, when we divide a by b we get an integer if and only if b | a. What can we say when b does not divide a? We can still make a statement using only integers by considering remainders. For example, we can say that 14 divided by 3 is 4 with a remainder of 2.

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

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