An officer of the army wants to arrange his soldiers to stand in rows of $12,15,16$ and $18$. If he wants to arrange them in a solid square also, then find the minimum number of soldiers.

To find the minimum number of soldiers, the number must be a multiple of $12,15,16,$ and $18$, and it must also be a perfect square (to form a solid square).
First, let's find the Least Common Multiple (LCM) of $12,15,16,$ and $18$ by prime factorization:

$$12=2^2\times 3^1$$

$$15=3^1\times 5^1$$

$$16=2^4$$

$$18=2^1\times 3^2$$

Taking the highest power of each prime factor:

* Highest power of $2$ is $2^4$
* Highest power of $3$ is $3^2$
* Highest power of $5$ is $5^1$

$$\text{LCM}=2^4\times 3^2\times 5^1=16\times 9\times 5=720$$

To make the total number of soldiers a perfect square, each prime factor in its factorization must have an even exponent.
Let's check the exponents in the LCM:

$$720=2^4\times 3^2\times 5^1$$

Here, the prime factors $2$ and $3$ have even powers ($4$ and $2$), but $5$ has an odd power ($1$). To make it a perfect square, we must multiply by $5$:

$$\text{Minimum number of soldiers}=720\times 5=3600$$

Since $3600=60^2$, it forms a perfect solid square. Thus, the minimum number of soldiers required is $3600$.