Finding the Least Perfect Square Divisible by Multiple Numbers

We need to find the least perfect square number that is divisible by 3, 4, 5, 6, and 8. For a number to be divisible by all these numbers, it must be a multiple of their Least Common Multiple (LCM).

Calculating the LCM of 3, 4, 5, 6, and 8

First, let's find the prime factorization of each number:

• 3 = 3¹
• 4 = 2²
• 5 = 5¹
• 6 = 2¹ × 3¹
• 8 = 2³

The LCM is found by taking the highest power of all prime factors that appear in any of the factorizations:

\(\text{LCM}(3, 4, 5, 6, 8) = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120\)

So, any number divisible by 3, 4, 5, 6, and 8 must be a multiple of 120.

Finding the Least Perfect Square Multiple of 120

A perfect square is a number where all the exponents in its prime factorization are even.

The prime factorization of the LCM, 120, is \(2^3 \times 3^1 \times 5^1\).

To make this a perfect square, we need to multiply it by factors that will make the exponents even. The current exponents are 3, 1, and 1. To make them even (the smallest even numbers greater than or equal to 3, 1, 1 are 4, 2, 2 respectively), we need to multiply by:

• For \(2^3\), we need \(2^1\) to get \(2^{3+1} = 2^4\).
• For \(3^1\), we need \(3^1\) to get \(3^{1+1} = 3^2\).
• For \(5^1\), we need \(5^1\) to get \(5^{1+1} = 5^2\).

The multiplying factor is \(2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30\).

The least perfect square divisible by 3, 4, 5, 6, 8 is the LCM multiplied by this factor:

\(\text{Least Perfect Square} = \text{LCM}(3, 4, 5, 6, 8) \times (2^1 \times 3^1 \times 5^1)\)

\(\text{Least Perfect Square} = 120 \times 30 = 3600\)

Verifying the Answer

Let's check if 3600 meets the conditions:

• Is 3600 a perfect square? Yes, \(60 \times 60 = 3600\).
• Is 3600 divisible by 3? \(3600 \div 3 = 1200\). Yes.
• Is 3600 divisible by 4? \(3600 \div 4 = 900\). Yes.
• Is 3600 divisible by 5? \(3600 \div 5 = 720\). Yes.
• Is 3600 divisible by 6? \(3600 \div 6 = 600\). Yes.
• Is 3600 divisible by 8? \(3600 \div 8 = 450\). Yes.

3600 is a perfect square and is divisible by all the given numbers. It is the least such number because we started with the LCM and multiplied by the smallest possible factors to make it a perfect square.

Looking at the options, 3600 is present and satisfies the conditions.