Quotient farthest mouths/nearest sources Much googling has not left me any the wiser, so here goes. Let FM be the distance between the mouths of two rivers X and Y. Let NS be the distance between the sources of the same two rivers X and Y. What pair of two rivers X and Y on Earth will produce the biggest quotient FM/NS? Or the smallest?

Two candidates for the biggest might be Indus and Brahmaputra but I am sure there are others.
Tbb is correct. The 'Congo and Nile' is by far the winner. Their mouths are 4600 km apart, whereas if you trek into the Rwenzori mountains, Western Uganda, there is s small highland swamp in a saddle point on the Congo-Uganda watershed. On either side there are several points where you can claim to be 'at the source', where you can stand with one foot in the Congo basin and the other foot in the Nile basin. The Indus and Brahmaputra are less than half this distance apart.

As for the other extreme, it's impossible to say. There are several rivers which end at more or less the same place. One then has to argue, 'Is one river a tributary of the other, or not?'