What is the the maximum possibe value of k for which 2013 can be written as sum of k consecutive positive integers ?

Answer

To find the maximum possible value of \( k \) for which 2013 can be written as the sum of \( k \) consecutive positive integers, we can follow these steps:

### Step 1: Set up the equation
Let the first integer in the sequence of \( k \) consecutive integers be \( n \). The integers can be expressed as:
\[ n, n+1, n+2, \ldots, n+k-1 \]
The sum \( S \) of these integers can be calculated using the formula for the sum of an arithmetic series:
\[ S = \frac{k}{2} \times (2n + (k - 1)) \]
This simplifies to:
\[ S = \frac{k}{2} \times (2n + k - 1) \]

### Step 2: Set the sum equal to 2013
We set the sum equal to 2013:
\[ \frac{k}{2} \times (2n + k - 1) = 2013 \]
Multiplying both sides by 2 gives:
\[ k(2n + k - 1) = 4026 \]

### Step 3: Factor 4026
Next, we need to find the factors of 4026. We can start by finding its prime factorization:
\[ 4026 = 2 \times 3 \times 11 \times 61 \]
From this, we can find all pairs of factors \( (k, m) \) such that \( k \times m = 4026 \).

### Step 4: List the factor pairs
The factor pairs of 4026 are:
- \( (1, 4026) \)
- \( (2, 2013) \)
- \( (3, 1342) \)
- \( (6, 671) \)
- \( (11, 366) \)
- \( (22, 183) \)
- \( (33, 122) \)
- \( (61, 66) \)

### Step 5: Determine valid values for \( k \)
For each factor \( k \), we need to check if \( n \) remains a positive integer. From the equation:
\[ 2n + k - 1 = \frac{4026}{k} \]
we can express \( n \) as:
\[ 2n = \frac{4026}{k} - k + 1 \]
\[ n = \frac{\frac{4026}{k} - k + 1}{2} \]
For \( n \) to be a positive integer, \( \frac{4026}{k} - k + 1 \) must be positive and even.

### Step 6: Check each factor
We will check each factor \( k \):
1. \( k = 1 \): \( n = \frac{4026 - 1 + 1}{2} = 2013 \) (valid)
2. \( k = 2 \): \( n = \frac{2013 - 2 + 1}{2} = 1006 \) (valid)
3. \( k = 3 \): \( n = \frac{1342 - 3 + 1}{2} = 670 \) (valid)
4. \( k = 6 \): \( n = \frac{671 - 6 + 1}{2} = 333 \) (valid)
5. \( k = 11 \): \( n = \frac{366 - 11 + 1}{2} = 178 \) (valid)
6. \( k = 22 \): \( n = \frac{183 - 22 + 1}{2} = 81 \) (valid)
7. \( k = 33 \): \( n = \frac{122 - 33 + 1}{2} = 45 \) (valid)
8. \( k = 61 \): \( n = \frac{66 - 61 + 1}{2} = 3 \) (valid)

### Step 7: Identify the maximum \( k \)
The maximum value of \( k \) from the valid factors is:
\[ \boxed{61} \]