Total number of factors of a greatest possible number which when divides 1313 and 621, the respective remainders obtained are 17 and 9 :

Options

A. 9
B. 10
C. 11
D. can't be determined

Answer

To solve the problem of finding the total number of factors of the greatest possible number that leaves remainders of 17 and 9 when dividing 1313 and 621 respectively, we can follow these steps:

### Step 1: Determine the numbers to be divided
We know that when dividing 1313 by the number \( x \), the remainder is 17. This means:
\[ 1313 - 17 = 1296 \]
Similarly, when dividing 621 by \( x \), the remainder is 9:
\[ 621 - 9 = 612 \]

### Step 2: Find the greatest possible number
The greatest possible number \( x \) that satisfies both conditions must be a divisor of both 1296 and 612. Therefore, we need to find the Highest Common Factor (HCF) of these two numbers.

### Step 3: Calculate the HCF of 1296 and 612
We can use the prime factorization method to find the HCF.

1. **Prime Factorization of 1296:**
- 1296 can be divided by 2:
- \( 1296 \div 2 = 648 \)
- \( 648 \div 2 = 324 \)
- \( 324 \div 2 = 162 \)
- \( 162 \div 2 = 81 \)
- 81 can be divided by 3:
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- Thus, the prime factorization of 1296 is:
\[ 1296 = 2^4 \times 3^4 \]

2. **Prime Factorization of 612:**
- 612 can be divided by 2:
- \( 612 \div 2 = 306 \)
- \( 306 \div 2 = 153 \)
- 153 can be divided by 3:
- \( 153 \div 3 = 51 \)
- \( 51 \div 3 = 17 \)
- Thus, the prime factorization of 612 is:
\[ 612 = 2^2 \times 3^2 \times 17^1 \]

3. **Finding the HCF:**
- The HCF is found by taking the lowest power of all prime factors present in both factorizations:
\[ \text{HCF} = 2^{\min(4,2)} \times 3^{\min(4,2)} = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 4: Calculate the total number of factors of the HCF
To find the total number of factors of 36, we use the formula for the number of factors based on its prime factorization:
- The prime factorization of 36 is:
\[ 36 = 2^2 \times 3^2 \]
- The number of factors can be calculated as:
\[ (2+1)(2+1) = 3 \times 3 = 9 \]

### Final Answer
The total number of factors of the greatest possible number is **9**.

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