How does the minimum wattage rating affect a UPS?

[Anonymous user]

I want to power a modem during power failure to keep the telephone line active.  No computer or other device would be connected to the UPS.  Just the modem.  The modem draws 6.5W (measured using one of those doo-dads you plug into the wall to measure wattage.)  The APC selection tool (awesome tool!!!) seems to suggest that this is too small a load for a UPS.  

I don't understand.  Does such a small load not mean the batteries would last longer or will a UPS not work at all for this case?

Any recommendations for a APC product that I should be looking at for this application?  (I already have an APC unit as backup to my main computer -- all I need it for there is orderly shutdown.)    In the case of the modem, I want to keep the phone line active as long as possible.  

[BillP]

@Anonymous user 

I use a BG500 for my modem/router. The UPS reports the device is pulling 14.1 watts. I also have a few BN675M1s that are powering lamps with LED bulbs incase the AC drops at night. Both will power the lamps with no issue. The bulbs pull around 9 watts.