Geography seems especially prone to problems like this, where the use of seemingly simple terms results in a problem that is too vague to answer without arbitrary criteria defining them, and the arbitrariness of the criteria renders the so-called "answer" meaningless.
'Source' and 'mouth' are too ill-defined to allow an answer that means anything to emerge. (And what about distance? As the geodetic crow flies, or along the river's course?)
Any two rivers whose drainage basins border each other can considered to have a 'source' at the same point, meaning NS =0, and the quotient, as defined by the problem, is infinite, no matter how distant the two rivers' 'mouths'.
Even if we select some arbitrary point as the 'source' we have the case of the Casiquiare and the Orinoco. The Casiquiare flows out from the Orinoco into the Rio Negro, a tributary of the Amazon. The source of the Casiquiare is the source of the Orinoco no matter where you put it. So, infinity again.
If two rivers share a delta, such as the Ganges and Brahmaputra or the Amazon and Tocatins, they have the same 'mouths' as far as anyone can define so their FM must be 0. But in that case their drainage basins border each other, so FM/NS = 0/0, which is the true answer,