Loser66
  • Loser66
Write gcd (210, 45) as a linear combination of 210 and 45 then use it to solve Diophantine 210x +45y = 60 Please, help
Mathematics
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
trantom
  • trantom
What does Diophantine mean?
Loser66
  • Loser66
My question is on second part. gcd (210, 45) =15 210 = 45(4) + 30 (a) 45= 30 + 15 (b) 30 = 15(2) +0 from (b) 15 = 45- 30 from (a) 30 = 210 +45(-4) hence 15 = 210 (-1) +45 (5) multiple both sides by 4, we get 60 = 210 (-4) +45 (20) hence \(x_0 = -4\\x = -4-3t ; t\in [0,14]\) \(y_0 = 20\\y = 20+14t; t\in[0,14]\)
trantom
  • trantom
I don't understand this yet sorry ganeshie can probably help :/

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Loser66
  • Loser66
But if I use (a) to solve, I have 30 = 210 +45(-4) then multiple both sides by 2 to get 60 = 210 (2) +45(-8) I have different answer with \(x_0 = -2 \\y_0=-8\) it confused me a lot.
Loser66
  • Loser66
sorry \(x_0 =2\)
ganeshie8
  • ganeshie8
Both are correct
Loser66
  • Loser66
What?? explain me, please
ganeshie8
  • ganeshie8
\((2, -8)\) and \((-4, 20)\) are both particular solutions to the diophantine equation \( 210x +45y = 60\) You just need "any one" particular solution to write out all the solutions.
ganeshie8
  • ganeshie8
for example, \((-1, 6)\) is another particular solution
Loser66
  • Loser66
What is the logic?
ganeshie8
  • ganeshie8
all the solutions = (any one particular solution) + (null solutions)
Loser66
  • Loser66
null solution is fixed, right? how particular one changes without changing the whole?
ganeshie8
  • ganeshie8
Let me ask you a question, whats the vector equation of the line that passes through \((-4,20)\) and has a direction vector \((-3,14)\) ?
Loser66
  • Loser66
You do scare me, hehehe... I think it is a vector parallel to vector (-3,14) through (-4,20)
ganeshie8
  • ganeshie8
write out the equation, im sure you will see the connection once you write out the equaiton
ganeshie8
  • ganeshie8
whats the vector equation of the line that passes through \((-4,20)\) and has a direction vector \((-3,14)\) ? we can express the line as \[\dbinom{x}{y} = \dbinom{-4}{20} + t\dbinom{-3}{14}\] yes ?
Loser66
  • Loser66
it is y = -14/3x -4/3
ganeshie8
  • ganeshie8
I have asked for equation in "vector" form
Loser66
  • Loser66
ok,
Loser66
  • Loser66
yes, I got this part
ganeshie8
  • ganeshie8
look at the vector equation that i wrote is that the only way to express the equation of line in vector form ?
ganeshie8
  • ganeshie8
How about below equation ? \[\dbinom{x}{y} = \dbinom{2}{-8} + t\dbinom{-3}{14}\] what line does this represent ? does it represent a different line ?
Loser66
  • Loser66
Yes, they are same.
ganeshie8
  • ganeshie8
how do you know they are same ?
Loser66
  • Loser66
just starting point is different, the first one starts at (-4,20), the second one at (2,-8) and both are on the same line.
ganeshie8
  • ganeshie8
Exactly! so do you agree that we can use "any one point" on the line to express the line in vector form ?
Loser66
  • Loser66
Yes, I do. :)
ganeshie8
  • ganeshie8
good, so do you see how your solution for diophantine equation and the vector form of a line are similar ?
Loser66
  • Loser66
Yes, I got it. Thank you so much. You are.... amazingly patient.
ganeshie8
  • ganeshie8
In short : 1) (-4, 20) is a `particular solution` vector 2) all the `null solutions` are given by : t(-3, 14) 3) therefore `all the solutions` are given by (-4, 20) + t(-3, 14)
Loser66
  • Loser66
My midterm is upcoming. I need practice more. Do you know some practice pages online I can use to work with?
ganeshie8
  • ganeshie8
which textbook are you using ?
Loser66
  • Loser66
Elementary number theory by Kenneth H. Rosen
ganeshie8
  • ganeshie8
I have david m burton if you want more practice problems on linear diophantine equations, i can take a screenshot of exercise page and send you..
Loser66
  • Loser66
Thank you so much. For this time, I focus on : 1) The fundamental theorem 2)Linear Diophantine 3)Congruences 4) Wilson's theorem , Fermat's theorem and Euler's theorem.
Loser66
  • Loser66
one more: Chinese theorem also
ganeshie8
  • ganeshie8
Here are some fun problems. https://s3.amazonaws.com/upload.screenshot.co/962c284bf3 Solving these on your own gives solid understanding of diophantine equations...
Loser66
  • Loser66
Yes, I will.
ganeshie8
  • ganeshie8
Few more https://s3.amazonaws.com/upload.screenshot.co/c46dd4303d
Loser66
  • Loser66
I will try all. Thanks a lot.

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