Question

Find the smallest square number which is exactly divisible by each of the number 5,8,12,15

Answer 1

We need to find the smallest square divisible by 5,8,12 and 15. This means that it will be the lowest square divisible by their LCM .
So let's first find the LCM of these numbers.
$5 = {5}^{1} \\ 8 = {2}^{3} \\ 12 = {2}^{2} \times 3 \\ 15 = 3 \times 5 \\ lcm = {2}^{3} \times 5 \times 3 = 120$
Now we have found that LCM of these numbers is 120 and
$120 = {2}^{3} \times 3 \times 5$
Now for a number to be perfect square, all the powers of its prime factors must be even. Thus the least perfect square divisible by 120 is
${2}^{4} \times {5}^{2} \times {3}^{2} = 3600$
Thus, the least square divisible by 5,8,12 and 15 is 3600 which is the square of 60.