Solve the diophantine equations: (2.a) 21x + 15y = 6 (2.b) 22u + 15v = 6

The particular solution of 2.a is and the complete solution is .

The particular solution of 2.b is and the complete solution is

Step-by-step explanation:

The Diophantine equation ax+by=n has solutions if and only if gcd(a,b) | n. If this is true, it has infinitely many solutions, and any solution can be used to generate a complete solution.

This are the steps that you need to follow:

So for the equation ,

When the remainder r = 0, the gcd is the divisor, 3, in the last equation so gcd(21,15) = 3

Then, observe that 3|6 (that means when we divide 6 by 3, the remainder is 0.) is true therefore there are integer solutions to the equation.

The same for the equation ,

When the remainder r = 0, the gcd is the divisor, 1, in the last equation so gcd(22,15) = 1 and 1|6 is true therefore there are integer solutions to this equation.

For the equation

For the equation

For the equation

This gives x_i=-2 and y_i=3 as a solution to the equation  

For the equation

This gives u_i=-2 and v_i=3 as a solution to the equation  

Then an initial solution to the equation is

For the equation

For this you can use this Theorem:

if  is an integer solution of the Diophantine equation ax+by=n, then all integer solutions to the equation are of the form

for some integer m.

The complete solutions for the equation are:

The complete solutions for the equation are:

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step-by-step explanation:

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