Question

Determine the smallest $3$-digit number which is exactly divisible by $6,8,\mathrm{and}12$.

Answer 1

Smallest three digits number.

First, we have to find the LCM of $6,8,\mathrm{and}12$.

Multiples of $6,8,\mathrm{and}12$ are,

Common Multiples of $6\to 6,12,18,\mathbf{24},30,36,42\dots ..$

Common Multiples of $8\to 8,16,\mathbf{24},32,40,48,54\dots \dots$

Common Multiples of $12\to 12,\mathbf{24},36,48,60,72,\dots$

It is observed that, $24$ is the least common multiple.

Thus, all the multiples of $24$ will also be divisible by $6,8,\mathrm{and}12$.

Now we will divide the smallest 3-digit number with the LCM obtained, and the remainder will be subtracted from the dividend, and $24$ will be added to it to make it perfectly divisible.

$\therefore \frac{100}{24}=4,\mathrm{Remainder}=4$

According to the above statement.

$$\begin{gathered} =(100–4)+24 \\ =96+24 \\ =120 \end{gathered}$$

Hence, the smallest 3-digit number which is exactly divisible by $6,8,\mathrm{and}12$ is $120$.