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## mathmath333 one year ago Find the number of integer solutions.

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1. mathmath333

\large \color{black}{\begin{align} 5x+8y=1\hspace{.33em}\\~\\ x<100,\ y<100 \end{align}}

2. anonymous

@satellite73 @ganeshie8

3. 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 $$2+5t\lt 100$$

4. anonymous

Hey, ganeshie, what branch of math is this? Number theory or something? Havent done a problem like it, so curious xD

5. 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

6. anonymous

Okay, heard of diophantine before. Ill take a look, thanks :)

7. ganeshie8

np :)

8. mathmath333

is the answer $$31$$ @ganeshie8

9. ganeshie8

$$-3-8t\lt 100 \implies t \gt -12.87$$ $$2+5t\lt 100 \implies t \lt 19.6$$ so $$-12.87 \lt t\lt 19.6$$

10. ganeshie8

that gives 32 solutions right ?

11. mathmath333

how did u count that

12. ganeshie8

-12.87 to 19.6 12 negative integers 19 positive integers and a zero

13. mathmath333

ok i forgot 0

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