Let's denote the total work as $W$.
Amal and Vimal together can do $W$ work in 150 days, so their combined efficiency is $\frac{W}{150}$. Vimal and Sunil together can do $W$ work in 100 days, so their combined efficiency is $\frac{W}{100}$.
Let $A$, $B$, and $C$ be the individual efficiencies of Amal, Vimal, and Sunil, respectively. From the given information, we can form the following equations:
1. $A + B = \frac{W}{150}$
2. $B + C = \frac{W}{100}$
Now, let's analyze the second scenario where they work on alternate days: Amal works on all days, so his work in one day is $A$. Vimal works on every second day, so his work in two days is $B$. Sunil works on every third day, so his work in three days is $C$.
So, in 6 days, they complete $A + B + C$ work.
To find the total number of days required, we need to find how many times 6 divides the total work $W$. We can calculate this by dividing $W$ by the work done in 6 days:
Total days = $\frac{W}{A + B + C}$
We can solve the equations for $A$, $B$, and $C$ and substitute them into the above equation to find the total number of days.
After solving, we get the total number of days as 139.
When $10^{100}$ is divided by 7, the remainder is ?