Find the least number which is exactly divisible by 10, 15, 20 and also a perfect square.

Given: 10, 15, 20

To find: We have to find the least number which is exactly divisible by 10, 15, 20 and also a perfect square.

Solution: 

First we need to find the LCM of the given numbers i.e. 10, 15 and 20.

Now,

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

Prime factorization of 10:

• 2 $\times $ 5 = 2¹ $\times $ 5¹

Prime factorization of 15:

• 3 $\times $ 5 = 3¹ $\times $ 5¹

Prime factorization of 20:

• 2 $\times $ 2 $\times $ 5 = 2² $\times $ 5¹

Highest power of each prime number:

• 2² , 3¹ , 5¹

Multiplying these values together:

2² $\times $ 3¹ $\times $ 5¹ = 60

Thus,

LCM(10, 15, 20) = 60

We know that in a perfect square all the prime factors of that number are in pairs. So, we need to multiply 60 with 3 and 5 to make it a perfect square.

60 $\times $ 3 $\times $ 5 = 900

So, the least number which is exactly divisible by 10, 15, 20 and also a perfect square is 900.