Prove that $$101x+37y=3819\cdots (1)$$ has a positive solution in integer.
*I tried solving this by applying division algorithm to $101$ & $37$
$$101=2(37)+27$$ Put this in eqn (1) will give $$27x + 37 (y\;+\;2x)=3819$$ Then taking $$x'=x, y'=y+2x$$ we have $$27x'+37y'=3819$$ Again applying division algorithm to $27$ & $37$, and following same procedure I got $$27x"+10y"=3819$$ where $$x"=x'+y'$$ & $$y"=y'$$ and repeating same procedure Similarly I got the step $$x^{(5)}+3y^{(5)}=3819$$ where $$x^{(5)}=x^{(4)}, y^{(5)}=y^{(4)}+2x^{(4)}$$.
Now if I apply division algorithm to $1$ & $3$ then remainder will be zero*
Now I don't know how to proceed from here. Kindly whoever knows about this method help me.
$3819=37\times 103+8$
Consider $101x+37(y-103)=8$.
\begin{align*} (37\times 2+27)x+37(y-103)&=8\\ 37(2x+y-103)+27x&=8\\ (27+10)(2x+y-103)+27x&=8\\ 27(3x+y-103)+10(2x+y-103)&=8\\ (10\times 2+7)(3x+y-103)+10(2x+y-103)&=8\\ 10(8x+3y-309)+7(3x+y-103)&=8\\ (7+3)(8x+3y-309)+7(3x+y-103)&=8\\ 7(11x+4y-412)+3(8x+3y-309)&=8\\ (3\times 2+1)(11x+4y-412)+3(8x+3y-309)&=8\\ 3(30x+11y-1133)+(11x+4y-412)&=8 \end{align*}
Set $30x+11y-1133=2$ and $11x+4y-412=2$. We have $x=14$ and $y=65$.
The general solution is $x=14+37n$, $y=65-101n$, where $n\in\mathbb{Z}$. $x$ and $y$ are both positive when $n=0$.
So, $x=14$ and $y=65$.
To reduce the number of steps, I will use negative "remainders" if its magnitude is smaller.
\begin{align*} (37\times 3-10)x+37(y-103)&=8\\ 37(3x+y-103)+10(-x)&=8\\ (10\times 4-3)(3x+y-103)+10(-x)&=8\\ 10(11x+4y-412)+3(-3x-y+103)&=8\\ (3\times 3+1)(11x+4y-412)+3(-3x-y+103)&=8\\ 3(30x+11y-1133)+(11x+4y-412)&=8 \end{align*}
With your work, you can take $x^{(5)}={3819}-37n$, $y^{(5)}=101n$. Then express $x$ and $y$ in terms of $n$ and find suitable value of $n$ to make both $x$ and $y$ positive.
The purpose of reducing $3819$ to $8$ is for easier observations.
Notice that $37\times 4=148$. So, $37(4)+10(-14)=8$. Compare with the equivalent form "$37(3x+y-103)+10(-x)=8$" of the equation, we see that $3x+y-103=4$ and $x=14$ give a solution.