Question

Find the least square number which is exactly divisible by $3,4,5,6$ and $8$.

Answer 1

Calculate the $\mathrm{LCM}$ of given numbers

Given numbers are: $3,4,5,6$ and $8$. Their $\mathrm{LCM}$ is,

$$\begin{gathered} \begin{array}{cccccc}2& 3& 4& 5& 6& 8\\ 2& 3& 2& 5& 3& 4\\ 2& 3& 1& 5& 3& 2\\ 3& 3& 1& 5& 3& 1\\ 5& 1& 1& 5& 1& 1\\ & 1& 1& 1& 1& 1\end{array} \\ \mathrm{LCM}\left(3,4,5,6,8\right)=2\times 2\times 2\times 3\times 5=120 \end{gathered}$$

To make it a perfect square, it must be multiplied by $2\times 3\times 5$ to get pairs of prime factors.

So, $120\times 30=3600$ is the least number which is exactly divisible by $3,4,5,6$ and $8$.

Hence, the required least square number is $3600$.