Find the least possible 5-digit number which, when divided by each of $12,10,16$ and $18$, leaves a remainder of $5$.

First, find the Least Common Multiple (LCM) of the divisors $12,10,16,$ and $18$:
* $12=2^2\times 3$
* $10=2\times 5$
* $16=2^4$
* $18=2\times 3^2$

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

The smallest 5-digit number is $10000$. Let's divide it by $720$ to find the nearest perfect multiple:

$$\frac{10000}{720}\approx 13.88$$

The smallest 5-digit multiple of $720$ is:

$$720\times 14=10080$$

To ensure a constant remainder of $5$, add $5$ to this multiple value:

$$\text{Required Number}=10080+5=10085$$