Write $\bar n$ for the decimal expansion of $n$ (with no leading 0). Let : be a symbol distinct from any digit. Let $a$ and $b$ be integers, with $a > 0$. Consider the language of solutions of the Diophantine equation $y=ax+b$:$$L = \{ \bar{x} \mathtt: \bar{y} \mid y = a\,x + b \}$$Is $L$ regular? context-free?(Contrast with Language of the values of an affine function)(Follows on How can solutions of a Diophantine equation be expressed as a language?) I think this would make a good homework question, so answers that start with a hint or two and explain not just how to solve the question but also how to decide what techniques to use would be appreciated. 
Language of the graph of an affine function
formal languages;regular languages;context free;integers
Hint:$53 \cdot 100000010000000000 + 4 = 5300000530000000004$.Sketch: Assume $a$ has $n$ digits and $b$ has $m$ digits. If $\overline{x}=10^k \underbrace{00\dots01}_{n}0^l\underbrace{00\dots0}_{m}$ for some $k,l$ then $\overline{ax+b}=\overline{a}0^k \overline{a} 0^l \overline{b}$. If $L$ was context-free, then  $$L'=L \cap 10^{\ast}0^{n-1}10^{\ast}0^m \mathtt: (0+1+2+...+9)^{\ast} = \{10^{k+n-1}10^{l+m} \mathtt: \overline{a}0^k\overline{a}0^l\overline{b}:k,l \in \mathbb N\}$$ would be context-free as well.  Given a PDA $M$ for $L'$ we could construct a PDA $N$ for $\{a^n b^m c^n d^m\}$ , a well-known non-context-free language. $N$ simulates $M$, but it is feeding different letters; it is a composition of $M$ with a transducer. Initially, $N$ behaves as if $M$ read $1$. Then, consecutive $a$'s are interpreted as $0$'s. Next, $N$ behaves as if $M$ read $0^{n-1}1$, and treats consecutive $b$'s as $0$'s, then as if it read $0^m \mathtt: \overline{a}$ etc.