Question

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

Answer 1

The least square number divisible by each of 3, 4, 5, 6 and 8 is equal to the LCM of 3, 4, 5, 6 and 8
$\begin{array}{cc}2& 3,4,5,6,8\\ 2& 3,2,5,3,4\\ 2& 3,1,5,3,2\\ 3& 3,1,5,3,1\\ 5& 1,1,5,1,1\\ & 1,1,1,1,1\end{array}$
$\therefore$ LCM of 3, 4, 5, 6 and 8 $=2\times 2\times 2\times 3\times 5=120$
The prime factorisation of 120 $=(2\times 2)\times 2\times 3\times 5$
Here, prime factors 2, 3 and 5 are unpaired. Clearly, to make it a perfect square, it must be multiplied by $2\times 3\times 5$ i.e. 30
Therefore, required number $=120\times 30=3600$
Hence, the least square number is 3600.