Amal and Vimal together can complete a task in 150 days, while Vimal and Sunil together can complete the same task in 100 days. Amal starts working on the task and works for 75 days, then Vimal takes over and works for 135 days. Finally, Sunil takes over and completes the remaining task in 45 days. If Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, then the number of days required to complete the task would have been:

The correct option is 139.

Let the work rates (amount of work done per day) of Amal, Vimal, and Sunil be $A$, $V$, and $S$ respectively. Let the total work to be completed be $1$ unit.

Based on the first set of given conditions, we can set up the following equations:
1. Amal and Vimal together can complete the task in 150 days:
$150(A+V)=1$
which simplifies to:
$A+V=\frac{1}{150}$
2. Vimal and Sunil together can complete the same task in 100 days:
$100(V+S)=1$
which simplifies to:
$V+S=\frac{1}{100}$

The next condition states: Amal works for 75 days, Vimal takes over and works for 135 days, and Sunil completes the remaining task in 45 days. The equation representing the completed work is:
$75A+135V+45S=1$
We can rewrite this expression to utilize our known terms $A+V$ and $V+S$:
$75(A+V)+60V+45S=1$
$75(A+V)+15V+45(V+S)=1$
Substitute the values of $A+V$ and $V+S$:
$75\left(\frac{1}{150}\right)+15V+45\left(\frac{1}{100}\right)=1$
$\frac{1}{2}+15V+\frac{9}{20}=1$
$15V=1-\frac{1}{2}-\frac{9}{20}$
$15V=\frac{1}{2}-\frac{9}{20}=\frac{1}{20}$
$V=\frac{1}{300}$

Now, we can find the individual rates of Amal and Sunil:
$A=\frac{1}{150}-V=\frac{2}{300}-\frac{1}{300}=\frac{1}{300}$
$S=\frac{1}{100}-V=\frac{3}{300}-\frac{1}{300}=\frac{2}{300}$

Under the new work schedule:
- Amal works on all days (Days: 1, 2, 3, 4, 5, 6, ...)
- Vimal works on every second day (Days: 2, 4, 6, ...)
- Sunil works on every third day (Days: 3, 6, ...)

This patterns repeats every 6 days. Let us calculate the work completed in a single 6-day cycle:
- Amal works for all 6 days: $6A$
- Vimal works for 3 days (days 2, 4, 6): $3V$
- Sunil works for 2 days (days 3, 6): $2S$
Total work done in a 6-day cycle:
$W_{\mathrm{cycle}}=6A+3V+2S$
Substitute the values of $A$, $V$, and $S$:
$W_{\mathrm{cycle}}=6\left(\frac{1}{300}\right)+3\left(\frac{1}{300}\right)+2\left(\frac{2}{300}\right)=\frac{6+3+4}{300}=\frac{13}{300}$

We want to find how many full 6-day cycles can be completed without exceeding 1 unit of work:
$n\times \frac{13}{300}\le 1\Rightarrow n\le \frac{300}{13}\approx 23.07$
Thus, 23 complete cycles of 6 days each are finished.
Total days for 23 cycles = $23\times 6=138$ days.
Work done in 138 days = $23\times \frac{13}{300}=\frac{299}{300}$ unit of work.

Remaining work left after 138 days is:
$W_{\mathrm{remaining}}=1-\frac{299}{300}=\frac{1}{300}$

Now, we proceed to Day 139, which is the 1st day of a new 6-day cycle. On the 1st day of the cycle:
- Amal works (since Amal works every day).
- Vimal does not work (works only on even days).
- Sunil does not work (works on days 3 and 6 of the cycle).
So, the work done on Day 139 is exactly Amal's daily rate, $A=\frac{1}{300}$.
This perfectly matches the remaining work of $\frac{1}{300}$, so the entire task is completed on Day 139.

Therefore, the number of days required to complete the task is 139.