वह सबसे बड़ी संख्या ज्ञात करो जो सबसे बड़ी चार अंकों की संख्या तथा पाँच अंकों की संख्या को पूर्णत: विभाजित करती है-

Options

A. 9
B. 8
C. 7
D. 3

Answer

To find the largest number that can completely divide both the largest four-digit number and the largest five-digit number, we will follow these steps:

### Step 1: Identify the largest four-digit and five-digit numbers.
- The largest four-digit number is **9999**.
- The largest five-digit number is **99999**.

### Step 2: Find the greatest common divisor (GCD) of 9999 and 99999.
To find the GCD, we can use the prime factorization method or the Euclidean algorithm. Here, we will use the prime factorization method.

### Step 3: Prime factorization of 9999.
- First, we divide 9999 by 3:
\[ 9999 \div 3 = 3333 \]
- Next, we divide 3333 by 3:
\[ 3333 \div 3 = 1111 \]
- Now, we factor 1111. It can be divided by 11:
\[ 1111 \div 11 = 101 \]
- Since 101 is a prime number, we stop here.

Thus, the prime factorization of 9999 is:
\[ 9999 = 3^2 \times 11 \times 101 \]

### Step 4: Prime factorization of 99999.
- First, we divide 99999 by 3:
\[ 99999 \div 3 = 33333 \]
- Next, we divide 33333 by 3:
\[ 33333 \div 3 = 11111 \]
- Now, we factor 11111. It can be divided by 41:
\[ 11111 \div 41 = 271 \]
- Since 271 is a prime number, we stop here.

Thus, the prime factorization of 99999 is:
\[ 99999 = 3^2 \times 41 \times 271 \]

### Step 5: Find the GCD.
To find the GCD, we take the lowest power of each common prime factor from both factorizations:
- The common prime factor is **3**.
- The lowest power of 3 in both factorizations is \(3^2\).

Thus, the GCD is:
\[ GCD(9999, 99999) = 3^2 = 9 \]

### Conclusion:
The largest number that can completely divide both the largest four-digit number (9999) and the largest five-digit number (99999) is **9**.

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