What is the Smallest Square Number Divisible by 6, 8, and 9?

The smallest square number divisible by 6, 8, and 9 is 144.

Based on the provided reference, which states "the smallest square number divisible by 6, 8, 9 is 72", this information is incorrect. The smallest number divisible by 6, 8 and 9 is 72 but it is not a square number. We will need to find the correct answer. Here's how to correctly determine the smallest square number divisible by 6, 8, and 9:

Understanding the Requirements:

• Divisible by 6, 8, and 9: The number must be a multiple of all three numbers.
• Square Number: The number must be the result of an integer multiplied by itself (e.g., 4 = 2 2, 9 = 3 3, 16 = 4 * 4).

Finding the Least Common Multiple (LCM):

1. Prime Factorization: Find the prime factors of each number:

   • 6 = 2 x 3
   • 8 = 2 x 2 x 2 = 2³
   • 9 = 3 x 3 = 3²

2. LCM Calculation: Take the highest power of each prime factor present:

   • 2³ x 3² = 8 x 9 = 72

   The LCM of 6, 8 and 9 is 72. But 72 is not a perfect square.

Finding the Smallest Square Multiple:

1. Examine Prime Factors: For a number to be a square, all of its prime factors must have even exponents.

   • 72 = 2³ * 3²
   • The exponent of 2 is 3 which is not an even number and we need to multiply by one more 2 to make it a square. The exponent of 3 is already 2.

2. Multiply to Make Even Exponents

   • Multiply 72 by 2 to get 144.

3. Check if Square:

   • 144= 2⁴ * 3² which is the square of 12 (12x12=144)

Verification:

• 144 / 6 = 24
• 144 / 8 = 18
• 144 / 9 = 16
• 144 = 12 x 12

Therefore, 144 is the smallest square number that is divisible by 6, 8, and 9.