The Greek Gift

The total work required is $180$ soldiers $\times 8$ days $= 1440$ soldier-days. Let the numbers of days the first $180$ soldiers worked, then $180 - 84= 96$ soldiers worked and $96 - 16 = 80$ soldiers worked be natural numbers $a$ , $b$ and $c$ respectively. Then, we have:

$\begin{aligned} 180a+96b+80c & = 1440 & \small \color{#3D99F6} \text{Dividing both sides by }80 \\ \frac {9a}4+\frac {6b}5+c & = 18 \end{aligned}$

Since the RHS is an integer, the LHS must also be an integer. This means that $4|a$ and $5|b$ . If $a=8$ , then $\dfrac {9a}4 = 18$ , $\implies b=c=0$ which is unacceptable. Therefore, $a=4$ . $\implies \dfrac {6b}5 + c = 9$ . Again, the only possible value is $b=5$ . $\implies c = 3$ .

Therefore, the total number of days needed is $a+b+c = 4+5+3= \boxed{12}$ .

Initially $180$ soldiers could finish the work in $8$ days, so each soldier could do $\dfrac{1}{8\cdot 180} = \dfrac{1}{1440}$ work in a day.

When $84$ soldiers left for war, $180-84 = 96$ soldiers remained, and they could do $\dfrac{96}{1440} = \dfrac{1}{15}$ work in a day.

When $16$ more fell sick, $96-16 = 80$ soldiers did the rest of the work. They could then do $\dfrac{80}{1440} = \dfrac{1}{18}$ work in a day.

Now let $x,y,z$ be the positive integers of workdays for the three scenarios. We can then create a Diophantine equation as the following:

$\dfrac{x}{8} + \dfrac{y}{15} + \dfrac{z}{18} = 1$

Thus, $45x + 24y + 20z = 360$ .

Since the end sum has an ending $0$ digit, $y$ needs to have a factor of $5$ . Otherwise, it can't end with zero.

However, if $y=10$ , then $120 = 45x + 20z$ or $24 = 9x + 4z$ . Then $24-4z = 4(6-z) = 9x$ , but $6-z$ is not a multiple of $9$ and $x$ can't be zero. And if $y=15$ , then $24y=360$ and $0=45x+20z$ , which is not applicable.

As a result, $y$ must be $5$ only.

Hence, $240 = 45x + 20z$ or $48 = 9x + 4z$ . Then $48-4z = 4(12-z) = 9x$ . The only integers that work are $z=3$ and $x=4$ .

Checking the answers:

$\dfrac{4}{8} + \dfrac{5}{15} + \dfrac{3}{18} = \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{6} = 1$

As a result, $180$ soldiers were working for $4$ days.

Then $96$ soldiers were working for $5$ days.

At last, $80$ soldiers were working for the last $3$ days and finished the work.

Finally, it would totally take $4+5+3 = \boxed{12}$ days to finish building the Greek gift.