Give the prime factorization of each of the following numbers :
9317

Answer

To find the prime factorization of the number 9317, we will follow these steps:

### Step 1: Check for divisibility by small prime numbers
We start by checking if 9317 is divisible by the smallest prime numbers: 2, 3, 5, 7, 11, etc.

- **Divisibility by 2**: 9317 is odd, so it is not divisible by 2.
- **Divisibility by 3**: The sum of the digits (9 + 3 + 1 + 7 = 20) is not divisible by 3, so 9317 is not divisible by 3.
- **Divisibility by 5**: The last digit is 7, so it is not divisible by 5.
- **Divisibility by 7**: When we divide 9317 by 7, we find that it is divisible.

### Step 2: Divide by 7
Now we divide 9317 by 7:

\[ 9317 \div 7 = 1331 \]

### Step 3: Factor 1331
Next, we need to factor 1331. We will again check for divisibility by small prime numbers.

- **Divisibility by 2**: 1331 is odd, so it is not divisible by 2.
- **Divisibility by 3**: The sum of the digits (1 + 3 + 3 + 1 = 8) is not divisible by 3, so 1331 is not divisible by 3.
- **Divisibility by 5**: The last digit is 1, so it is not divisible by 5.
- **Divisibility by 11**: When we divide 1331 by 11, we find that it is divisible.

### Step 4: Divide by 11
Now we divide 1331 by 11:

\[ 1331 \div 11 = 121 \]

### Step 5: Factor 121
Next, we factor 121. We can see that:

\[ 121 = 11 \times 11 = 11^2 \]

### Step 6: Combine all factors
Now we can combine all the factors we found:

- From Step 2: \( 9317 = 7 \times 1331 \)
- From Step 4: \( 1331 = 11 \times 121 \)
- From Step 5: \( 121 = 11^2 \)

Thus, we have:

\[ 9317 = 7 \times 11 \times 11 \times 11 = 7 \times 11^3 \]

### Final Answer
The prime factorization of 9317 is:

\[ 9317 = 7 \times 11^3 \]

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