A boy plays a mathematical game where he tries to write the number 1998 into the sum of 2 or more consecutive positive even numbers (e.g., 1998 = 998 + 1000). In how many different ways can he do so?

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem of expressing the number 1998 as the sum of two or more consecutive positive even numbers, we can follow these steps:

### Step 1: Understand the Sum of Consecutive Even Numbers
The sum of \( n \) consecutive even numbers starting from \( 2k \) can be expressed as:
\[ S = 2k + (2k + 2) + (2k + 4) + \ldots + (2k + 2(n-1)) \]
This simplifies to:
\[ S = n(2k + (n-1)) = n(2k + n - 1) \]
Thus, we can express the sum \( S \) as:
\[ S = n(2k + n - 1) \]

### Step 2: Set Up the Equation
We want to find \( n \) and \( k \) such that:
\[ n(2k + n - 1) = 1998 \]
This means \( 2k + n - 1 = \frac{1998}{n} \).

### Step 3: Factor 1998
To find possible values of \( n \), we need to factor 1998. The prime factorization of 1998 is:
\[ 1998 = 2 \times 3^3 \times 37 \]
Now, we can find the divisors of 1998.

### Step 4: List the Divisors
The divisors of 1998 can be calculated as follows:
- The factors of 1998 are: \( 1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666, 999, 1998 \).

### Step 5: Find Valid \( n \)
We need \( n \) to be a divisor of 1998 such that \( n \geq 2 \) (since we need at least two consecutive even numbers). The valid divisors are:
- \( 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666, 999 \).

### Step 6: Calculate \( k \) for Each \( n \)
For each valid \( n \), calculate \( k \):
\[ 2k = \frac{1998}{n} - n + 1 \]
Thus,
\[ k = \frac{\frac{1998}{n} - n + 1}{2} \]
We need \( k \) to be a positive integer.

### Step 7: Check Each Valid \( n \)
Now we will check each valid \( n \):
1. For \( n = 2 \): \( k = \frac{999 - 2 + 1}{2} = 499 \) (valid)
2. For \( n = 3 \): \( k = \frac{666 - 3 + 1}{2} = 332 \) (valid)
3. For \( n = 6 \): \( k = \frac{333 - 6 + 1}{2} = 164 \) (valid)
4. For \( n = 9 \): \( k = \frac{222 - 9 + 1}{2} = 107 \) (valid)
5. For \( n = 18 \): \( k = \frac{111 - 18 + 1}{2} = 47 \) (valid)
6. For \( n = 37 \): \( k = \frac{54 - 37 + 1}{2} = 9 \) (valid)
7. For \( n = 74 \): \( k = \frac{27 - 74 + 1}{2} = -23.5 \) (not valid)
8. For \( n = 111 \): \( k = \frac{18 - 111 + 1}{2} = -46 \) (not valid)
9. For \( n = 222 \): \( k = \frac{9 - 222 + 1}{2} = -106 \) (not valid)
10. For \( n = 333 \): \( k = \frac{6 - 333 + 1}{2} = -163 \) (not valid)
11. For \( n = 666 \): \( k = \frac{3 - 666 + 1}{2} = -331 \) (not valid)
12. For \( n = 999 \): \( k = \frac{2 - 999 + 1}{2} = -498 \) (not valid)

### Step 8: Count the Valid Combinations
The valid values of \( n \) that yield a positive integer \( k \) are \( 2, 3, 6, 9, 18, 37 \). Thus, there are 6 different ways to express 1998 as the sum of two or more consecutive positive even numbers.

### Final Answer
The boy can write the number 1998 into the sum of two or more consecutive positive even numbers in **6 different ways**.

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