Question

Find the smallest square number that is divisible by each of the numbers $8,15,$

and $20.$

Answer 1

The number that is perfectly divisible by each of the numbers $8,15,$ and $20$ is their LCM.

$\begin{array}{cccc}2& 8& 15& 20\\ 2& 4& 15& 10\\ 2& 2& 15& 5\\ 3& 1& 15& 5\\ 5& 1& 5& 5\\ & 1& 1& 1\end{array}$

LCM of $8,15,$ and $20=\underset{–}{2\times 2}\times 2\times 3\times 5=120$

Here, prime factors $2,3,$ and $5$ do not have their respective pairs. Therefore, $120$ is not a perfect square.

Therefore, $120$ should be multiplied by $2\times 3\times 5,$ i.e. $30,$ to obtain a perfect square.

Hence, the required square number is $120\times 2\times 3\times 5=3600$, which is divisible by each of the numbers $8,15,$ and $20.$