Which number did this member say in the first round? 96 members of a counting club are standing in a large circle. They start saying numbers 1, 2, 3, etc. in turn,
going around the circle. Every member that says an even number steps out of the circle
and the rest continue, starting the second round with 97. They continue in this way until only one member is
left. Which number did this member say in the first round?
(A) 1 (B) 17 (C) 33 (D) 65 (E) 95
I think A but correct answer is 65 , how?  and what the easy way? (I tried count one by one , its too long... there must be easy way!)
 A: The first person is going to remain in the game until the very end because (1) there are an even number of people at the beginning, and (2) an even number of people step out after the first five rounds.  (The number eliminated after the first five rounds are $48, 24, 12, 6,$ and $3,$ respectively.)
In other words, $1$ says an odd number as long as the previous round ended with an even number of people.
When three people remain, they will be distributed evenly: $1, 33, 65$.  Member $33$ is eliminated after the sixth round, and member $1$ after the seventh.
A: Let's assume each member gets a label with the first number they say.
In the first round, those with labels congruent to $0 \pmod 2$ leave the room.
$48$ people have left the room at this point and the last number was $96$.
In the second round, those with labels congruent to $3\pmod 4$ leave the room.
$72$ people have left the room at this point and the last number was $144$.
In the third round, those with labels congruent to $5\pmod 8$ leave the room.
$84$ people have left the room at this point and the last number was $168$.
In the fourth round, those with labels congruent to $9 \pmod 16$ leave the room.
$90$ people have left the room at this point and the last number was $180$.
In the fifth round, those with labels congruent to $17 \pmod 32$ leave the room.
$93$ people have left the room at this point and the last number was $186$.
In the sixth round, those with labels congruent to $33 \pmod 64$ leave the room.
$94$ people have left the room at this point and the last number was $189$.
Now the seventh round starts with only two players: $1$ and $65$. Player $1$ says $190$ and leaves. So the winner is player $65$.
