In this task, you need to provide the correct option for a given problem from the provided options.

Example Input: Problem: if x and y are both odd prime numbers and x < y , how many distinct positive integer q factors does 2 xy have ? 
Options: a ) 3 , b ) 4 , c ) 6 , d ) 8 , e ) 12
Example Output: d

Example Input: Problem: a bottle contains a certain solution . in the bottled solution , the ratio of water to soap is 3 : 4 , and the ratio of soap to salt is five times this ratio . the solution is poured into an open container , and after some time , the ratio of water to soap in the open container is halved by water evaporation . at that time , what is the ratio of water to salt in the solution ? 
Options: a ) 9 : 32 , b ) 9 : 30 , c ) 32 : 9 , d ) 19 : 32 , e ) 18 : 30
Example Output: a

Example Input: Problem: set a consists of the integers from 4 to 16 , inclusive , while set b consists of the integers from 6 to 20 , inclusive . how many distinct integers do belong to the both sets at the same time ? 
Options: a ) 5 , b ) 7 , c ) 8 , d ) 9 , e ) 11
Example Output:
e