By which smallest number should 20184 be multiplied so that it becomes a perfect square ?

Options

A. 2
B. 3
C. 5
D. 6

Answer

To determine the smallest number by which 20184 should be multiplied to become a perfect square, we will follow these steps:

### Step 1: Prime Factorization of 20184
We begin by finding the prime factorization of 20184.

1. Since 20184 is even, we can divide it by 2:
- \( 20184 \div 2 = 10092 \)
2. Continuing to divide by 2:
- \( 10092 \div 2 = 5046 \)
- \( 5046 \div 2 = 2523 \)

3. Now, 2523 is not even, so we check for divisibility by the next prime number, which is 3:
- \( 2523 \div 3 = 841 \)

4. Next, we factor 841. We check for divisibility by prime numbers:
- \( 841 = 29 \times 29 \) (since \( 29^2 = 841 \))

Thus, the complete prime factorization of 20184 is:
\[ 20184 = 2^3 \times 3^1 \times 29^2 \]

### Step 2: Identify the Exponents
In the prime factorization, we have:
- \( 2^3 \)
- \( 3^1 \)
- \( 29^2 \)

For a number to be a perfect square, all the exponents in its prime factorization must be even.

### Step 3: Adjust the Exponents
Now, we check the exponents:
- The exponent of 2 is 3 (odd), so we need one more 2 to make it even (3 + 1 = 4).
- The exponent of 3 is 1 (odd), so we need one more 3 to make it even (1 + 1 = 2).
- The exponent of 29 is 2 (even), so it is already fine.

### Step 4: Calculate the Smallest Number to Multiply
To make the number a perfect square, we need to multiply by:
- One more 2: \( 2^1 \)
- One more 3: \( 3^1 \)

Thus, the smallest number we need to multiply by is:
\[ 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Conclusion
Therefore, the smallest number by which 20184 should be multiplied to become a perfect square is **6**.

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