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Here's the idea: given $w$ and $n$, we find a number $v$ such that $vw \equiv 1 \pmod{n}$. In your case of $w=7$ and $n=173$, we have $v=99$; such a $v$ will always exist if $w$ and $n$ are relatively prime, and can be found by the extended Euclidean algorithm. The value $v$ allows us to map from the "hard values" to the "superincreasing ones". Here's how ...



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