Question

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

Answer 1

Answer:

The particular solution of 2.a is $x_{i}=-4, y_{i}=6$ and the complete solution is $x=-4+5m, y=6-7m$.

The particular solution of 2.b is $u_{i}=-12, v_{i}=18$ and the complete solution is $u=-12+15m, v=18-22m$

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:

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

So for the equation $21x+15y=6$,

$21=1*15+6\\15=2*6+3\\6=2*3+0$

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 $22u+15v=6$,

$22=1*15+7\\15=2*7+1\\7=7*1+0$

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 $21x+15y=6$

$6=21-1*15\\3=15-2*6$

For the equation $22u+15v=6$

$1=15-2*7\\7=22-1*15$

• 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 $21x_{i}+15y_{i}=3$

$3=15-2*6\\3=15-2(21-1*15)\\3=(3*15)+(-2*21)\\3=21*(-2)+15*3$

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

For the equation $22u+15v=1$

$1=15-2*7\\1=15-2*(22-1*15)\\1=3*15+(-2*22)\\1=22*(-2)+15*3$

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

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

Then an initial solution to the equation $21x+15y=3$ is

$x_{i}=(-2*\frac{6}{3})=-4\\y_{i}=(3*\frac{6}{3})=6$

For the equation $22u+15v=1$

$u_{i}=(-2*\frac{6}{1})=-12\\v_{i}=(3*\frac{6}{1})=18$

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

For this you can use this Theorem:

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

$(x_{i} +m\frac{b}{gcd(a,b)} ,y_{i} -m\frac{a}{gcd(a,b)})$

for some integer m.

The complete solutions for the equation $21x+15y=3$ are:

$(-4 +m\frac{15}{3} ,6 -m\frac{21}{3})$

The complete solutions for the equation $22u+15v=1$ are:

$(-12 +m\frac{15}{1} ,18 -m\frac{22}{1})$