Two consecutive multiples of a certain number add upto 184. The number is-

Options

A. 4
B. 8
C. 23
D. 46

Answer

To solve the problem, we need to find a certain number \( X \) such that the sum of two consecutive multiples of \( X \) equals 184. Let's break this down step by step.

### Step 1: Define the consecutive multiples
Let the first multiple of the number \( X \) be \( nX \) (where \( n \) is a positive integer). The next consecutive multiple will be \( (n + 1)X \).

### Step 2: Set up the equation
According to the problem, the sum of these two consecutive multiples is 184:
\[ nX + (n + 1)X = 184 \]

### Step 3: Simplify the equation
Combine the terms on the left side:
\[ nX + nX + X = 184 \]
\[ (2n + 1)X = 184 \]

### Step 4: Solve for \( X \)
Now, we can express \( X \) in terms of \( n \):
\[ X = \frac{184}{2n + 1} \]

### Step 5: Analyze the factors of 184
To find suitable values for \( n \), we need \( 2n + 1 \) to be a factor of 184. Let's find the factors of 184:
- The prime factorization of 184 is \( 2^3 \times 23 \).
- The factors of 184 are: 1, 2, 4, 8, 23, 46, 92, and 184.

### Step 6: Check which factors can be expressed as \( 2n + 1 \)
Since \( 2n + 1 \) must be odd (as it represents an odd number), we only consider the odd factors of 184:
- The odd factors of 184 are: 1 and 23.

### Step 7: Calculate \( X \) for each odd factor
1. If \( 2n + 1 = 1 \):
- \( 2n = 0 \) → \( n = 0 \)
- \( X = \frac{184}{1} = 184 \) (not a valid multiple since we need consecutive multiples)

2. If \( 2n + 1 = 23 \):
- \( 2n = 22 \) → \( n = 11 \)
- \( X = \frac{184}{23} = 8 \)

### Conclusion
The number \( X \) is 8. Therefore, the two consecutive multiples of 8 are \( 88 \) (11 × 8) and \( 96 \) (12 × 8), and their sum is indeed 184.

### Final Answer
The number is \( \boxed{8} \).