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The problem is from Borevich and Shafarevich's number theory.

Show that the equation $a_1x_1 + \cdots +a_nx_n = b$, where $a_1,\ldots,a_n,b$ are integers, is solvable if and only if the corresponding congruence is solvable for all values of the modulus $m$.

Frankly, I am at a loss of how to approach the $\impliedby$ direction. I would appreciate some hints to get me started in the right direction.

Kirk Bonney
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Hint: by Bezout it is solvable in $\Bbb Z \color{#c00}\iff d:=\gcd(a_1,\ldots,a_n)\mid b.\ $ If it's solvable for all moduli then it's solvable $\bmod d,\,$ so there are $\,x_i$ with $\bmod d\!:\,\ b \equiv a_1x_1 + \cdots +a_nx_n\color{#0a0}{\equiv 0},\,$ by $\,\color{#0a0}{d\mid a_i}.\,$ Thus $\,d\mid b\ \ \color{#c00}{\rm thus}$ it is solvable in $\,\Bbb Z.$

Bill Dubuque
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