Find the least number which when divided by $4,6,8,10$ and $12$ leaves the remainder $3$ in all cases.

The required least number is given by:
$$\text{Required Number}=\text{LCM}(4,6,8,10,12)+\text{Remainder}$$

First, let's find the $\text{LCM}$ of $4,6,8,10,\text{ and }12$:

* $4=2^2$
* $6=2\times 3$
* $8=2^3$
* $10=2\times 5$
* $12=2^2\times 3$

$$\text{LCM}=2^3\times 3\times 5=8\times 3\times 5=120$$

Adding the required common remainder ($3$):

$$\text{Required Number}=120+3=123$$