Find the smallest natural number by which 980 should be multiplied to make it a perfect square.

Options

A. 7
B. 5
C. 3
D. 6

Answer

To find the smallest natural number by which 980 should be multiplied to make it a perfect square, we will follow these steps:

### Step 1: Prime Factorization of 980
First, we need to factorize 980 into its prime factors.

- Start dividing by the smallest prime number, which is 2:
- \( 980 \div 2 = 490 \)
- \( 490 \div 2 = 245 \)
- Now, we can divide 245 by the next smallest prime number, which is 5:
- \( 245 \div 5 = 49 \)
- Finally, we can factor 49:
- \( 49 \div 7 = 7 \)
- \( 7 \div 7 = 1 \)

So, the prime factorization of 980 is:
\[ 980 = 2^2 \times 5^1 \times 7^2 \]

### Step 2: Identify the Exponents
Next, we look at the exponents of the prime factors:
- For \( 2^2 \), the exponent is 2 (which is even).
- For \( 5^1 \), the exponent is 1 (which is odd).
- For \( 7^2 \), the exponent is 2 (which is even).

### Step 3: Determine the Missing Factors
To make 980 a perfect square, all the exponents in its prime factorization must be even.

- The exponent of 5 is odd (1). To make it even, we need to multiply by \( 5^1 \) (which is 5).

### Step 4: Multiply by the Required Factor
Thus, we multiply 980 by 5:
\[ 980 \times 5 = 4900 \]

### Step 5: Verify if the Result is a Perfect Square
Now, let's check if 4900 is a perfect square:
- The prime factorization of 4900 is:
\[ 4900 = 2^2 \times 5^2 \times 7^2 \]
- All exponents are even, confirming that 4900 is indeed a perfect square.

### Final Answer
The smallest natural number by which 980 should be multiplied to make it a perfect square is:
\[ \boxed{5} \]

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