Abhi remembers that the birthday is after  13^th September but before 16^th September   , while Rakesh remembers that birthday is after 14^th September  but before 18^th September . On which date of September is the birthday?

15^th September 

According to question , 

According to Abhi  birthday is after 13^th September but before 16^th September 

Possible date of birthday= 14 , 15      ------ ( 1 ) 

According to Rakesh birthday is after 14^th September  but before 18^th September 

Possible date of birthday = 15 , 16 , 17      ----- (2 ) 

from ( 1 ) & ( 2) 

Common satisfying date is 15^thSeptember 

So birthday is on 15^thSeptember