In these types of questions generally, we have to make the FA with min. state which is equal to the LCM of both the boundary ie.LCM(3,5) this is because we have to take care of different states and cases for 3 the strings must be stopped at the 6th, 9th, 12th &15th state whereas, in the case of 5, the string must be stopped at the 11th, 16th (1st state) and so on.
So, after the 15th state, a pattern is formed. so, min. no. of states is 15 needed.
Hope you understand