Finding the Least Square Number

The problem asks for the smallest perfect square number that is exactly divisible by 8, 12, and 20.

To find this, we first need to calculate the Least Common Multiple (LCM) of 8, 12, and 20. Then, we'll adjust the LCM to make it a perfect square.

Calculate the LCM

Find the prime factorization of each number:

• $8 = 2^3$
• $12 = 2^2 \times 3^1$
• $20 = 2^2 \times 5^1$

The LCM is found by taking the highest power of each prime factor present in any of the numbers:

LCM = $2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120$

Making the LCM a Perfect Square

The LCM is 120. Its prime factorization is $2^3 \times 3^1 \times 5^1$. For a number to be a perfect square, all the exponents in its prime factorization must be even.

Currently, the exponents are 3, 1, and 1. We need to increase them to the next highest even numbers (which would be 4, 2, and 2 respectively) by multiplying by the necessary factors.

• To make the exponent of 2 even (from 3 to 4), we need to multiply by $2^1$.
• To make the exponent of 3 even (from 1 to 2), we need to multiply by $3^1$.
• To make the exponent of 5 even (from 1 to 2), we need to multiply by $5^1$.

The minimum factor needed to make the LCM a perfect square is $2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30$.

Final Calculation

Multiply the LCM by this factor:

Least Square Number = LCM $\times$ Factor = $120 \times 30 = 3600$

Alternatively, build the perfect square using the required prime factors:

Required Square Number = $2^4 \times 3^2 \times 5^2 = 16 \times 9 \times 25 = 3600$

The number 3600 is a perfect square ($60^2$) and is divisible by 8, 12, and 20.

Conclusion: The least square number divisible by 8, 12, and 20 is 3600.