The sum of all natural number belonging to the set `{1,2,3. . . 100}`, whose `H.C.F` with `2304` is ____

Options

A. 2449
B. 1633
C. 1449
D. 2633

Answer

To solve the problem of finding the sum of all natural numbers from the set {1, 2, 3, ..., 100} whose H.C.F with 2304 is a certain value, we will follow these steps:

### Step 1: Understanding H.C.F and the Problem
The H.C.F (Highest Common Factor) of a number with 2304 will determine which numbers we need to consider. We need to find the natural numbers from 1 to 100 that have a specific H.C.F with 2304.

### Step 2: Factorizing 2304
First, we factor 2304 to understand its prime factors:
\[ 2304 = 2^8 \times 3^2 \]

### Step 3: Identifying the Conditions
For a number \( n \) to have a specific H.C.F with 2304, it must be a multiple of the factors of 2304. The possible H.C.F values can be formed by taking combinations of the prime factors of 2304.

### Step 4: Finding the Relevant Multiples
We need to find all the multiples of each factor of 2304 that are less than or equal to 100. The relevant multiples will depend on the specific H.C.F we are looking for.

### Step 5: Calculating the Sum of Relevant Numbers
Once we identify the numbers that meet the H.C.F condition, we will sum them up. The sum of the first \( n \) natural numbers can be calculated using the formula:
\[ S = \frac{n(n + 1)}{2} \]
where \( n \) is the largest number in the set.

### Step 6: Example Calculation
For example, if we want to find the sum of all natural numbers whose H.C.F with 2304 is 16, we would find the multiples of 16 up to 100:
- The multiples of 16 are: 16, 32, 48, 64, 80, 96.

Now, we sum these numbers:
\[ 16 + 32 + 48 + 64 + 80 + 96 = 336 \]

### Step 7: Conclusion
The final answer will depend on the specific H.C.F value we are looking for. If we take H.C.F = 16 as an example, the sum would be 336.