I'm looking for help on solving Linear Diophantine Equation
15y + 27x = 39
y = (39-27x)/15
y = (39 - 12x - 15x)/15
y = (39 - 12x)/15 - x
Next step? I'm unsure
Thanks in advance
I'm not sure this is how you are expected to solve this, or even if thisOriginally Posted by delpin
is the simplest method; but:
First divide through by and rearrange to get:
Now this gives:
So there exists an integer such that:
Now the LHS (left hand side) is even, so must be odd.
So let , then:
But the LHS is divisible by so the RHS must also be divisible by
so must be even., so we may write it as , and then:
Now substituting this back into gives:
So we see that for any :
is a solution.
Use the following theorem,
Given a diophatine equation with
and is a particular solution
Then all solutions and every solution is,
for an integer
You can leave it the way it is but I suggest to divide by 3,
You need to find a specific solution. Which can be done with trail and error. (Otherwise you can use the Euclidean Algorithm or Coutinued fractions to get a particular solution).
We can easily see that and work.
Also which divides 13.
Thus, all solutions are,
Note it might look different from CaptainBlack's but they are both equivalent.
This is basically Captain Black's and Hacker's solutions ... in baby-talk.
First, reduce the equation: .I'm looking for help on solving Linear Diophantine Equation:
We have: .
. . Then: .
Multiply both sides by 4:
Hence: . for some integer
Substitute into the original equation: .
. . and we get: .
The solutions are: . for any integer