Question

Find all the nonnegative integer solutions to each Diophantine equation

12x + 57y = 423

Answer 1

There are no nonnegative integer solutions to Diophantine equation 12x + 57y = 423

We have Diophantine equation 12x + 57 y = 423

We try to find the nonnegative integer values of x and y.

Now, first find gcd

gcd(12, 57) = 3.

And 3 divides 423.

So, the equation is solvable.

The given equation in reduced form is 4x+19y=141

We found by the continued fraction method above

4(3)-13(1)=-1

Multiplying by -c=-141

4(3)(-141)-13(1)(-141)=141

so a particular solution is (-423,-141)

Adding, we get

4(19t)+19(-4t)=0

The general solution is: (x, y)=(19t-423,-141-4t)

For non-negative solutions

$19t\geq 423,-141\geq 4t \\t\geq \frac{423}{19} ,t\leq \frac{-141}{4}$

As the value of t cannot be found.

So, there are no positive solution for the given equation.

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