If you attempt to fit the following set of numbers into a standard interlocking crossword-style digit grid, which of the numbers will NOT fit logically?

In standard grid puzzles of this type, numbers are fitted based on their common intersecting digits.
* Grouping the digits shows that the numbers $137$, $301$, $731$, and $770$ can form mutually intersecting patterns across columns and rows because they share the recurring terminal/starting numbers $1$, $3$, $7$, and $0$.
* For example, $137$ (horizontal) connects seamlessly with $731$ or $770$ at its last digit.
* Analysis of the structural intersections reveals that the term 173 does not possess a consistent digit arrangement that allows it to interlock within this specific numerical set, leaving it out.