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

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:

Use the Euclidean algorithm to compute gcd(a,b)=d

So for the equation [tex]21x+15y=6[/tex],

[tex]21=1*15+6\\15=2*6+3\\6=2*3+0[/tex]

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 [tex]22u+15v=6[/tex],

[tex]22=1*15+7\\15=2*7+1\\7=7*1+0[/tex]

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.

The next step is reformat the equations from the Euclidean algorithm as follows:

For the equation [tex]21x+15y=6[/tex]

[tex]6=21-1*15\\3=15-2*6[/tex]

For the equation [tex]22u+15v=6[/tex]

[tex]1=15-2*7\\7=22-1*15[/tex]

Using substitution, go through the steps of the Euclidean algorithm to find a solution to the equation ax_{i}+by_{i}=d

For the equation [tex]21x_{i}+15y_{i}=3[/tex]

[tex]3=15-2*6\\3=15-2(21-1*15)\\3=(3*15)+(-2*21)\\3=21*(-2)+15*3[/tex]

This gives x_i=-2 and y_i=3 as a solution to the equation [tex]21x_{i}+15y_{i}=3[/tex]

For the equation [tex]22u+15v=1[/tex]

[tex]1=15-2*7\\1=15-2*(22-1*15)\\1=3*15+(-2*22)\\1=22*(-2)+15*3[/tex]

This gives u_i=-2 and v_i=3 as a solution to the equation [tex]22u_{i}+15v_{i}=1[/tex]

The initial solution to the equation ax+by=n is the ordered pair [tex](x_{i}*\frac{n}{d},y_{i}*\frac{n}{d} )[/tex]

Then an initial solution to the equation [tex]21x+15y=3[/tex] is

[tex]x_{i}=(-2*\frac{6}{3})=-4\\y_{i}=(3*\frac{6}{3})=6[/tex]

For the equation [tex]22u+15v=1[/tex]

[tex]u_{i}=(-2*\frac{6}{1})=-12\\v_{i}=(3*\frac{6}{1})=18[/tex]

Find the complete solutions of the equation ax+by=n

For this you can use this Theorem:

if [tex](x_{i},y_{i})[/tex] is an integer solution of the Diophantine equation ax+by=n, then all integer solutions to the equation are of the form

[tex](x_{i} +m\frac{b}{gcd(a,b)} ,y_{i} -m\frac{a}{gcd(a,b)})[/tex]

for some integer m.

The complete solutions for the equation [tex]21x+15y=3[/tex] are:

[tex](-4 +m\frac{15}{3} ,6 -m\frac{21}{3})[/tex]

The complete solutions for the equation [tex]22u+15v=1[/tex] are:

[tex](-12 +m\frac{15}{1} ,18 -m\frac{22}{1})[/tex]