Find the least number by which 294 must be multiplied to make it a perfect square.

Question

Accepted Answer

To determine the least number by which 294 must be multiplied to make it a perfect square, we need to analyze the prime factorization of 294.

1. **Prime Factorization of 294:**
   - 294 is an even number, so it is divisible by 2:  
     $ 294 \div 2 = 147 $
   - Next, 147 is divisible by 3 (since the sum of its digits, 1 + 4 + 7 = 12, is divisible by 3):  
     $ 147 \div 3 = 49 $
   - 49 is a perfect square, as it is $ 7 \times 7 $.

Therefore, the prime factorization of 294 is:  
   $ 294 = 2 \times 3 \times 7 \times 7 $.

2. **Making 294 a Perfect Square:**
   - For a number to be a perfect square, all prime factors must appear in pairs.
   - In the factorization of 294, the prime factors are: 2, 3, and 7 (with 7 already paired).
   - The factor 2 appears once, and the factor 3 appears once. To make them pairs, we need one more 2 and one more 3.

3. **Calculating the Least Multiplier:**
   - To pair the unpaired 2 and 3, we need to multiply 294 by $ 2 \times 3 = 6 $.

4. **Verification:**
   - Multiplying 294 by 6 gives:  
     $ 294 \times 6 = 1764 $.
   - The prime factorization of 1764 is:  
     $ 1764 = 2^2 \times 3^2 \times 7^2 $, which confirms it is a perfect square.

5. **Explanation of Other Options:**
   - **Option 1 (3):** Multiplying by 3 alone would not pair the factor 2, so it would not result in a perfect square.
   - **Option 2 (4):** Multiplying by 4 would pair the factor 2 but not the factor 3, so it would not result in a perfect square.
   - **Option 4 (12):** Multiplying by 12 would indeed make it a perfect square, but it is not the least number required. The least number is 6.

Therefore, the least number by which 294 must be multiplied to make it a perfect square is indeed 6, supporting Option 3 as the correct answer.