Diophantine equations for integers.

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Solve 172x + 20y 1000• Then all solutions are x x o + (b/d)ty y o –(a/d)t where t is an integer.andFrom the equation 172x + 20y 1000,we see that a 172 and b 20.From our previous work, x o 500,y o - 4250, and d 4.So the solutions, in integers, arex 500 + 5t and y - 4250 – 43twhere t ranges over the integers.
ExampleFind all positive solutions to theDiophantine equation 172x + 20y 1000.we need to find those values of t for whichx 500 + 5t > 0 and y - 4250 – 43t > 0.
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: Theorem 4If the linear Diophantine
Page 27: Find all positive integer solutions

Diophantine equations for integers.

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