Is it ok to answer only the day of the month to the question "今天几号?"? Is it ok to answer only the day of the month to the question "今天几号?"?

First dialog:

-- 今天几号?

-- 今天是八月二十号. (it is OK, common phrase)

Second dialog:

-- 今天几号?

-- 今天是二十号. (is it Ok? Why not?)

If the second dialog is not grammaticaly correct, how can I ask for a day of the month?

UPDATE: 

It is ok to use month when asking 你的生日是几月几号? But the date (month + day) is asked without month (from book 汉语会话301句) just 今天几号?
Yes, it totally makes sense to answer only the day since the question is asking for the day only instead of month and day.

If one is asking for month it would be something like: 今天是几月几号？