Prove that there exists $c_1, ... c_n$ such that $a_1c_1 + ... + a_nc_n = b$ iff $a_1x_1 + ... + a_nx_n\equiv b \;(\bmod\; m)$ is solvable

Question

($n, m$ are natural numbers, and $a_1,...a_n,c_1, ... c_n, b$ are integers)

I'm quite lost on how to approach this problem, any help would be appreciated.

Answer 1

Presumably it means "solveable $\color{#0a0}{\text{for all }\, m}$" (or all $\,m\ge k\,$ or all $\,m\in S),\,$ else it is clearly false.

$\begin{align}{\rm Notice}\ \ \bmod m\!:\,\ &\exists\, x_i\!:\ a_1 x_1 +\cdots+ a_n x_n \equiv b\!\!\pmod{\!m}\\[.2em] \iff\ &\exists\, x_i\!:\ a_1 x_1 +\cdots+ a_n x_n + m x_ 0 = b\\[.2em] \iff\ &\ \ \,\gcd(a_1,\ldots,\color{#c00}{a_n,m})\mid b,\ \ \text{by Bezout} \end{align}$

and, similarly, notice $\,\gcd(a_1,\ldots,a_n)\mid b\,$ is the solvability condition for your first equation.

Thus it suffices to choose $\,m\in S\,$ such that the two gcds are equal (e.g. if $\,\color{#c00}{a_n\mid m})$. Presumably your specific set $S$ is defined such that it contains a nonzero multiple of some $\,a_i,\,$ enabling you to choose such an $\,m\,$ that equalizes the gcds, which is clearly true if $\,\color{#0a0}{S=\Bbb N}\,$ or all $\,m\ge k.$

Answer 2

The “only if” part is straightforward, but I don’t think the “if” part is true if $m$ is a single natural number. $2x=3 \mod 5$ is solvable but there is no integer $c$ such that $2c=5$.