# 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.

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