Diophantine equations for integers.

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Greatest Common Divisord is the greatest common divisor of<strong>integers</strong> a and b if d is the largest integerwhich is a common divisor of both a and b.Notation: d gcd(a, b)Example: ±2, ±7, and ±14 are the only<strong>integers</strong> that are common divisors of both42 and 56. Since 14 is the largest,gcd(42, 56) 14.
Use of the gcdReducing fractionsEx.42 143 3 56 144 4However: not all fractions are easily reduced!Ex.80518633
Page 1: The Euclidean Algorithm andDiophant
Page 5 and 6: Euclidean Algorithm(p. 102)To find
Page 7 and 8: Theorem(3.2.2, p.105)For any nonzer
Page 9 and 10: • Solve each division problem, ex
Page 11 and 12: Ex) Now use the Euclidean Algorithm
Page 13 and 14: Theorem 1For any nonzero integers a
Page 15 and 16: Theorem 4If gcd(a,b)│c, then the
Page 17 and 18: Using Division and Euclidean Algori
Page 19 and 20: • Use the Euclidean Algorithm to
Page 21 and 22: Solve 172x + 20y 10004 2 172 (1 7
Page 23 and 24: Theorem 4If the linear Diophantine
Page 25 and 26: ExampleFind all positive solutions
Page 27: Find all positive integer solutions

Diophantine equations for integers.

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