Diophantine equations

1.  Solve the equations:

 ax + by = 0  and  ax  + by = c

Solution:

The Greek mathematician DIOPHANTUS  (about 250 BC) dealt with solving equations of the type ax + by = c, where only integer solutions were accepted. Such equations are called “Diophantine equations”.

 

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2.In the set Z solve the equation:

6x – 4y = 0

Solution:

diophantine-equations-2.gif

3.In the set Z solve the equation:

3x + 15y = 0

Solution:

diophantine-equations-3.gif

4.Find out which of the following equations are solvable in the set Z:

diophantine-equations-4z.gif
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5.In the set Z solve the equation:

10x + 4y = 16
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6.In the set Z solve the equation:

21x +15y = 3
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7.What integer dimensions can a rectangle have if its perimeter is 24 cm?

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8.In what ways can you pay the sum of 21 euros if you only have 2-euro and 5-euro coins?

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9.Determine how many ways 22 liters of wine can be transferred into 2-liter and 3-liter containers.

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10.A ribbon 50 cm long is to be cut into pieces 6 cm and 4 cm long. In how many ways can this be done?

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11.Determine all isosceles triangles whose side lengths are integers and whose perimeter equals 40 cm.

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