Find the number of integer solutions.

Question

mathmath333 · Answer

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

anonymous · Answer

@satellite73 @ganeshie8

ganeshie8 · Answer

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 $2+5t\lt 100$

anonymous · Answer

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

ganeshie8 · Answer

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 · Answer

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

ganeshie8 · Answer

np :)

mathmath333 · Answer

is the answer $31$  @ganeshie8

ganeshie8 · Answer

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

ganeshie8 · Answer

that gives 32 solutions right ?

mathmath333 · Answer

how did u count that

ganeshie8 · Answer

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

mathmath333 · Answer

ok i forgot 0