For admission to various affiliated colleges, a university conducts a written test with four different sections,each with a maximum of 50 marks. The following table gives the aggregate as well as the sectional cut-offmarks fixed by six different colleges affiliated to the university. A student will get admission only if he/she gets marks greater than or equal to the cut-off marks in each of the sections and his/her aggregate marks are at least equal to the aggregate cut-off marks as specified by the college.
| College | Cutoff |
|---|---|
| A | 170 |
| B | 180 |
| C | 192 |
| D | 196 |
| Option | Possible Status |
|---|---|
| 181 | Potential |
| 176 | Potential |
| 184 | Potential but does not exceed threshold |
| 196 | Unlikely, as it surpasses lower cut-offs |
| 190 | Possible, yet slightly higher |





For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: