Diophantine equations for integers.

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Find all positive integer solutions to the equation172x + 20y 1000.All solutions are x 500 + 5t and y - 4250 – 43t.x 500 + 5t > 0 t > - 100.y - 4250 – 43t > 0 t < - 98.83…Since t must be an integer, t -99.So - 100 < t - 99.
Find all positive integer solutions to the equation172x + 20y 1000.All solutions are x 500 + 5t and y - 4250 – 43t.We just found that - 100 < t - 99.Since t must be an integer,- 100 < t - 99 t -99. So there isonly one positive solution to the<strong>Diophantine</strong> equation, namelyx 500 + 5t ··· 5 andy - 4250 – 43t ··· 7.
Page 1 and 2: The Euclidean Algorithm andDiophant
Page 3 and 4: Use of the gcdReducing fractionsEx.
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: ExampleFind all positive solutions

Diophantine equations for integers.

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