Finding the Smallest Perfect Square Divisible by 8 and 12

To find the smallest perfect square that is divisible by both 8 and 12, we first need to understand what it means for a number to be divisible by both 8 and 12. A number that is divisible by two or more numbers is a common multiple of those numbers. The smallest such number is the Least Common Multiple (LCM).

Step 1: Find the LCM of 8 and 12

We find the LCM by using the prime factorization method:

• Prime factorization of 8: \(8 = 2 \times 2 \times 2 = 2^3\)
• Prime factorization of 12: \(12 = 2 \times 2 \times 3 = 2^2 \times 3^1\)

To find the LCM, we take the highest power of all prime factors present in the factorizations of 8 and 12.

• The prime factors are 2 and 3.
• Highest power of 2: \(2^3\) (from 8)
• Highest power of 3: \(3^1\) (from 12)

So, the LCM of 8 and 12 is \(2^3 \times 3^1 = 8 \times 3 = 24\).

This means any number divisible by both 8 and 12 must be a multiple of 24 (e.g., 24, 48, 72, 96, 120, 144, ...).

Step 2: Find the Smallest Perfect Square Multiple of 24

Now we need to find the smallest multiple of 24 that is also a perfect square. A perfect square is a number that can be obtained by squaring an integer (e.g., \(1^2=1\), \(2^2=4\), \(3^2=9\), \(12^2=144\), etc.).

In terms of prime factorization, a number is a perfect square if and only if all the exponents in its prime factorization are even.

The prime factorization of 24 is \(2^3 \times 3^1\). To make this a perfect square, we need to multiply it by the smallest possible factors that will make all exponents even. The exponent of 2 is 3 (odd), and the exponent of 3 is 1 (odd).

• To make the exponent of 2 even, we need at least one more factor of 2 (exponent 3 + 1 = 4).
• To make the exponent of 3 even, we need at least one more factor of 3 (exponent 1 + 1 = 2).

So, we need to multiply 24 by \(2^1 \times 3^1 = 6\).

The smallest multiple of 24 that is a perfect square is \(24 \times 6\).

\(24 \times 6 = 144\).

Step 3: Verify the Answer

Let's check if 144 meets the criteria:

• Is 144 a perfect square? Yes, \(144 = 12^2\).
• Is 144 divisible by 8? Yes, \(144 \div 8 = 18\).
• Is 144 divisible by 12? Yes, \(144 \div 12 = 12\).

Since 144 is a perfect square and is divisible by both 8 and 12, and it is the smallest such multiple of 24, it is the smallest perfect square divisible by both 8 and 12.

Analysis of Options

Let's quickly look at the provided options:

Based on the analysis, only 144 is a perfect square that is divisible by both 8 and 12. Among the given options, it is the smallest that satisfies the conditions.

Revision Table: Key Concepts

Additional Information: Perfect Squares and Multiples

Understanding perfect squares and multiples is fundamental in number theory. When a question asks for a number divisible by two or more numbers, the concept of LCM is central. When it adds the constraint of being a perfect square, you need to combine the ideas.

Any multiple of a number 'n' can be written as \(n \times k\) for some integer \(k\). If we want this multiple to be a perfect square, its prime factorization must have all exponents being even. We achieve this by identifying the prime factors in 'n' that have odd exponents and multiplying by the minimum necessary factors to make them even. The smallest such multiplier is unique.

For example, to find the smallest perfect square multiple of 50:

• Prime factorization of 50: \(50 = 2 \times 5 \times 5 = 2^1 \times 5^2\).
• The exponent of 2 is 1 (odd), and the exponent of 5 is 2 (even).
• To make the exponent of 2 even, we need to multiply by \(2^1\).
• The smallest multiplier is 2.
• The smallest perfect square multiple of 50 is \(50 \times 2 = 100\). \(100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2\), which has even exponents.

This method is general and applies to finding the smallest perfect square divisible by any set of integers by first finding their LCM and then adjusting its prime factorization.