Damn, I didn't notice from your OP that you were using bipolar 74LS series. In that case your 10k is about right, but pulled
UP to 5V rather than down to GND. Bipolar LS TTL logic has very unequal high and low currents (Input : about 20uA high and 360uA low, Output: 400
uA high and 8
mA low).
It is much harder to pull an LS TTL input low with a resistor, you need to achieve no more than 0.8V with the 360uA input current flowing. If you pull up then you only need to get above 2.7V with 20uA flowing. If you try to use pull-down resistors then your other gates will have difficulty overcoming them with their relatively weak output high capability. Floating (unconnected) bipolar TTL inputs are an old trap, they float high on their own but are very sensitive to noise.
In the same way, if you want to drive an LED from LS TTL, then you put it (and its current limiting resistor) between the output and +5V. This means that it lights on logic low.
This may a get a bit confusing for a young mind (it made my head spin a bit just typing it!). I really suggest that you consider using CMOS 74HC logic instead. Its input current is symmetrical at <1uA (hence 100k resistor) and the outputs are also symmetrical at +/- 4mA. It
is more static sensitive, but not as much as modern low voltage CMOS. Its 2 - 6V VCC and very low supply current make it much more battery friendly too.
Sorry for the confusion. I've linked a 74LS00 datasheet and 74HC00 datasheet for reference - they are representative of their respective logic family electrical parameters.
http://ecelabs.njit.edu/student_resources/datas/74ls00.pdfhttp://pdf.datasheetcatalog.com/datasheet/on_semiconductor2/74HC00.PDF