61. One hundred students, each with a number on their T-shirt, stood in a circle in the following sequence of their numbers: $1,2,3, \ldots, 100$. On command, they started to leave: the one with number 1 on the T-shirt stayed, the one with number 2 left, the one with number 3 stayed, the one with number 4 left, and so on, every other one around the circle until only one student remained. What number is on the T-shirt of this student?
61. When those with even numbers leave, 50 students with odd numbers remain: $-1,3,5, \ldots, 99$. Next, those with numbers $3,7,11, \ldots$ leave, after which 25 students remain. The last remaining number is 97, since $1+4 \cdot(25-1)=1+96=97$, i.e., the remaining numbers are $1,5,9,13, \ldots, 97$. Next, numbers $5,13, \ldots$ leave. In this process, 12 students leave, the last of whom has the number 93, since $5+8 \cdot(12-1)=93$, and 13 remain. Their numbers are:

$$
1,9,17,25,33,41,49,57,65,73,81,89,97
$$

Since the last one to leave had the number 93, number 97 remains, and then $1,17,33,49,65,81,97$ leave. After this, $25,57,89$ leave, followed by the number 41 and, finally, 9. The number 73 remains.