What is the smallest square number which is divisible by 4, 6, and 32?

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Accepted Answer

To find the smallest square number divisible by 4, 6, and 32:

1. **Find the LCM of the numbers:**  
   - Prime factorization:  
     $$ 
     4 = 2^2, \quad 6 = 2 	imes 3, \quad 32 = 2^5 
     $$  
   - LCM is obtained by taking the highest powers of all prime factors:  
     $$ 
     	ext{LCM} = 2^5 	imes 3 = 96.
     $$

2. **Ensure it is a perfect square:**  
   - For a number to be a perfect square, all prime factors must appear in even powers. In the LCM (96), the power of $$ 2 $$ is already even ($$ 2^5 $$), but the power of $$ 3 $$ is odd ($$ 3^1 $$).  
   - Multiply by an additional $$ 3 $$ to make the power of $$ 3 $$ even and by an additional $$ 2 $$ to make the power of $$ 2 $$ even:  
     $$ 
     	ext{Smallest square number} = 96 	imes 2 	imes 3 = 576.
     $$

Thus, the smallest square number divisible by 4, 6, and 32 is **576**, which corresponds to option **(d)**.

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