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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.