Question

Find the smallest square number that is divisible by each of the numbers $4,9$ and $10$.

Answer 1

Now, to find a number which is divisible by $4,9$ and $10$, we have to find its LCM first.

$\begin{array}{cc}2& 4,9,10\\ 2& 2,9,5\\ 3& 1,9,5\\ 3& 1,3,5\\ 5& 1,1,5\\ & 1,1,1\end{array}$

L.C.M. of $4,9$ and $10$ is $180$.

Now, using prime factorization,

$\begin{array}{cc}2& 180\\ 2& 90\\ 3& 45\\ 3& 15\\ 5& 5\\ & 1\end{array}$

$\Rightarrow 180=2\times 2\times 3\times 3\times 5$

$\Rightarrow 180=2^2\times 3^2\times 5$

As, $5$ does not have any pair, therefore, $180$ must be multiplied by $5$ to make it a perfect square.

$\Rightarrow 180\times 5=2^2\times 3^2\times 5^2=900$

Hence, the smallest square number which is divisible by $4,9$ and $10$ is $900$.