I am self studying number theory from Introduction to Analytic number theory by Apostol.
I have a doubt in one argument of proof of Theorem 5.14 and I am posting its image.
My doubt is in 2nd line of 2 nd paragraph how Apostol writes - If y is a solution of (5) then ay $\equiv $ at ( mod m) . How does y being a solution of (5) implies that?
Can someone please help
$t\,$ is a root of $(7),$ which, scaled by $d,\,$ shows $t$ is root of $(5)$, so $\,at\equiv b\equiv ay\Rightarrow at\equiv ay,\,$ by transitivity of congruence (being an equivalence relation).
The proof is poorly presented. Below is a more conceptual presentation that generalizes widely. Like linear differential and difference equations (recurrences), for any linear congruence $\,ax \equiv b,\,$ it's easy to show (see here or here) the $\rm\color{darkorange}{general}$ solution is the sum of any $\color{#0a0}{{\rm particular\ solution}\ x_0}\,$ plus the general solution of associated homogeneous equation $ax \equiv 0\pmod{\!m},\,$ which here is $\, m\mid ax\!\iff m/d\mid (a/d)x\!\iff m/d\mid x,\,$ by Euclid & $\,(m/d,a/d)\! =\! (m,a)/d = 1.\,$ Viewed $\!\bmod m\ $ such $\:\!x\:\!$ are the $\,\color{#c00}d\,$ multiples of $\,m/d,\,$ i.e. $\, m/d\,\color{#c00}{\{0,1,2,\ldots, d\!-\!1\}}^{\phantom{|^|}}\!$ [to be rigorous note $\!\bmod m\!:\,\ x\,\equiv\, k(m/d)\,\equiv\, k(m/d)\bmod m\:\overset{\small\rm\color{#90f}{DL}}\equiv\: m/d\,\color{#c00}{(k\bmod d)}$ by ${\small\rm\color{#90f}{DL}}\!=\!$ $\!\bmod\!$ Distrib. Law].
Hence the $\rm\color{darkorange}{general}$ solution is $\!\bmod m\!:\, x\equiv \!\!\!\!\!\!\!\!\underbrace{\color{#0a0}{x_0} + m/d\ \color{#c00}k^{\phantom{|^{|^.}}\!\!\!}}_{\text{$\rm \color{#0a0}{particular}$ + homogeneous}}\!\!\!\!\!\!\!\!\!,\ \color{#c00}{0\!\le\! k\!<\! d}\ \ $ [$x_0 = t\,$ in the OP]
Remark $ $ Said structure of the solution space will be clarified when one studies linear algebra and modules (= linear algebra where the coefficient algebra is a ring vs. field)
See here for more on switching back-and-forth between the equivalent linear congruence and its associated Bezout linear equation, and links to handy algorithms for solving such equations.
See here for a convenient view of the above in terms of modular (multi-valued) fractions