Find the smallest number by which `980` be multiplied so that the product is a perfect square.

Options

A. `11`
B. `4`
C. `8`
D. `5`

Answer

To find the smallest number by which 980 must be multiplied to make it a perfect square, we will use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 980
We start by dividing 980 by the smallest prime number, which is 2.

- **Divide by 2**:
\[ 980 \div 2 = 490 \]

- **Divide by 2 again**:
\[ 490 \div 2 = 245 \]

- **Now divide by 5** (next prime number):
\[ 245 \div 5 = 49 \]

- **Now divide by 7**:
\[ 49 \div 7 = 7 \]

- **Finally, divide by 7 again**:
\[ 7 \div 7 = 1 \]

Now, we can write the prime factorization of 980:
\[ 980 = 2^2 \times 5^1 \times 7^2 \]

### Step 2: Analyze the Prime Factors
To form a perfect square, all prime factors must have even powers.

- The prime factorization we found is:
- \(2^2\) (even)
- \(5^1\) (odd)
- \(7^2\) (even)

### Step 3: Identify the Unpaired Prime Factor
From the analysis:
- The factor \(2^2\) is already paired.
- The factor \(7^2\) is also paired.
- The factor \(5^1\) is unpaired (odd).

### Step 4: Find the Smallest Number to Multiply
To make the power of \(5\) even, we need to multiply by \(5\) (to make it \(5^2\)).

Thus, the smallest number by which 980 should be multiplied to make it a perfect square is:
\[ \text{Smallest number} = 5 \]

### Conclusion
Therefore, the answer is:
\[ \text{The smallest number by which 980 should be multiplied is } 5. \]

---