I am trying problems from Apostol Modular functions and Dirichlet series in number theory and I could not think about this problem from chapter 2 .
Problem is – Given integers $a, b, c, d\;$ with $ad-bc \equiv 1 \pmod n$, prove that there always exists integers $\alpha,\beta, \gamma, \delta$ such that $\alpha \equiv a \pmod n$, $\beta \equiv b \pmod n$, $\gamma \equiv c \pmod n$, $\delta \equiv d \pmod n$ with $\alpha \delta-\beta \gamma = 1$ .
I am unable to think how to prove existence of $\alpha, \beta, \gamma, \delta$ which are equivalent to $a, b, c, d \bmod n$ respectively. Can someone please give hints.
$(\overbrace{a\!+\!\ell n}^{\textstyle \alpha},\:\!b)=1\,$ for $\,\ell\in\Bbb Z\,$ by $\,{(a,b,n)=1,}\,$ by here. Let $\,\beta =b.\,$ We solve for $\,\delta,\gamma\in\Bbb Z$.
$\!\!\bmod n\!:\ \color{#0a0}{\alpha\equiv a}\,\Rightarrow \color{#0a0}\alpha d\!-\!b c = \color{#0a0}ad\!-\!bc = 1,\ $ so $\,\alpha d\! -\! b c = \color{#c00}{1\!-\!kn}\,$ for $\,k\in\Bbb Z,\,$ hence
$1\! =\! \underbrace{\alpha(d\!+\!in)\!-\!b(c\!+\!jn)}_{\textstyle \alpha\,\delta \,-\, \beta\,\gamma}\! =\! \color{#c00}{1\!-\!kn}\!+\!(i\alpha\!-\!jb)n\!\iff\! i\alpha\!-\!jb = k.\,$ Such $\,i,j\,$ exist by $(\alpha,b)\!=\!1$