Problem:

Finding the day of the week matching the last day of each month (serial number) listed in column A.

Solution:

Use the WEEKDAY and DATE functions as shown in the following formula:
=WEEKDAY(DATE(2004,A2+1,0),2)
Thus, the formula will return a number between 1 & 7 for each month in column A, where 1 represents Monday, and 7 represents Sunday.
To return 1 for Sunday and 7 for Saturday, modify your formula as follows:
=WEEKDAY(DATE(2004,A2+1,0),1)