The least number by which 216 must be divided to make the result a perfect square is

Options

A. 3
B. 4
C. 6
D. 9

Answer

To find the least number by which 216 must be divided to make the result a perfect square, we can follow these steps:

### Step 1: Factor the number 216
First, we need to find the prime factorization of 216. We can do this by dividing 216 by the smallest prime numbers.

- Divide by 2:
\( 216 \div 2 = 108 \)
- Divide by 2 again:
\( 108 \div 2 = 54 \)
- Divide by 2 again:
\( 54 \div 2 = 27 \)
- Divide by 3:
\( 27 \div 3 = 9 \)
- Divide by 3 again:
\( 9 \div 3 = 3 \)
- Divide by 3 again:
\( 3 \div 3 = 1 \)

So, the prime factorization of 216 is:
\[ 216 = 2^3 \times 3^3 \]

### Step 2: Identify the powers of the prime factors
From the prime factorization \( 2^3 \times 3^3 \), we can see the powers of the prime factors:
- The power of 2 is 3.
- The power of 3 is 3.

### Step 3: Determine the necessary adjustments to make it a perfect square
For a number to be a perfect square, all the powers in its prime factorization must be even.

- The power of 2 (which is 3) is odd. To make it even, we need to reduce it by 1 (from 3 to 2). This means we need to divide by \( 2^1 \).
- The power of 3 (which is 3) is also odd. To make it even, we need to reduce it by 1 (from 3 to 2). This means we need to divide by \( 3^1 \).

### Step 4: Calculate the least number to divide
Now, we need to find the product of the factors we need to divide by:
\[ 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Conclusion
Thus, the least number by which 216 must be divided to make the result a perfect square is **6**.

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