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There is a relation between rectilinear-adjacent squares such that there is a unique rectilinear path from the top-left corner of the grid down to the bottom-right corner of the grid. Each square can participate in the path only once. What is the relation and the path it induces?

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I think the relation is

not coprime, i.e. having a common divisor > 1

Example:

31 is connected to 93 because gcd(93,31) = 31 > 1
93 is connected to 9 because gcd(93, 9) = 3 > 1
93 is not connected to 56 because gcd(93, 56) = 1

And the path is:

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