Find the least square number which is exactly divisible by 8, 9 and 10

Given: 

8, 9, 10

To find: 

We have to find the least square number which is exactly divisible by 8, 9 and 10.

Solution: 

First, we need to find the LCM of the given numbers i.e., 8, 9 and 10.

Now,

Writing all the numbers as a product of their prime factors:

Prime factorization of 8:

• $2\times2\times2 = 2^3$

Prime factorization of 9:

• $3\times 3 = 3^2$

Prime factorization of 10:

• $2\times5 = 2^1\times 5^1$

Highest power of each prime number:

• $2^3 , 3^2 , 5^1$

Multiplying these values together:

$2^3\times 3^2\times5^1 = 360$

Thus,

LCM $(8, 9, 10) = 360$

We know that in a perfect square all the prime factors of that number are in pairs. So, we need to multiply 360 with 2 and 5 to make it a perfect square.

$360\times 2\times 5 = 3600$

So, the least square number which is exactly divisible by 8, 9 and 10 is 3600.