mathmath333
  • mathmath333
Find the number of integer solutions.
Mathematics
katieb
  • katieb
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mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} 5x+8y=1\hspace{.33em}\\~\\ x<100,\ y<100 \end{align}}\)
anonymous
  • anonymous
ganeshie8
  • ganeshie8
By inspection \((-3,2)\) is one solution, and the null solution is \((-8t,5t)\) Therefore all the solutions are given by \[(-3,2)+(-8t,5t)\] which is same as \[(-3-8t,~2+5t)\] so we need to find the number of \(t\) values such that \(-3-8t\lt 100\) and \(2+5t\lt 100\)

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anonymous
  • anonymous
Hey, ganeshie, what branch of math is this? Number theory or something? Havent done a problem like it, so curious xD
ganeshie8
  • ganeshie8
yes.. linear diophantine equations... here is a much simpler problem http://math.stackexchange.com/questions/897356/how-to-find-natural-solutions-of-an-equation/897369#897369
anonymous
  • anonymous
Okay, heard of diophantine before. Ill take a look, thanks :)
ganeshie8
  • ganeshie8
np :)
mathmath333
  • mathmath333
is the answer \(31\) @ganeshie8
ganeshie8
  • ganeshie8
\(-3-8t\lt 100 \implies t \gt -12.87\) \(2+5t\lt 100 \implies t \lt 19.6\) so \(-12.87 \lt t\lt 19.6\)
ganeshie8
  • ganeshie8
that gives 32 solutions right ?
mathmath333
  • mathmath333
how did u count that
ganeshie8
  • ganeshie8
-12.87 to 19.6 12 negative integers 19 positive integers and a zero
mathmath333
  • mathmath333
ok i forgot 0

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