Problem 5. The number 585 can be written as the sum of two consecutive integers, 292 and 293. What is the greatest number of consecutive positive integers whose sum is 585?

A. 13
B. 30
C. 34
D. 39
E. 15



Answer :

To find the greatest number of consecutive positive integers whose sum is 585, we can follow these steps:

1. Understand the problem:
- We need to express 585 as the sum of the maximum number of consecutive positive integers.

2. Formulate the sum of consecutive integers:
- Suppose we have \(k\) consecutive integers starting from \(x\).
- The integers would be \(x, x+1, x+2, \ldots, x+(k-1)\).
- The sum \(S\) of these \(k\) integers can be expressed as:
[tex]\[ S = x + (x+1) + (x+2) + \ldots + (x+(k-1)) \][/tex]

3. Simplify the sum:
- This sum can be written as:
[tex]\[ S = xk + (0 + 1 + 2 + \ldots + (k-1)) \][/tex]
- The sum of the first \(k\) integers is \(\frac{k(k-1)}{2}\).
- Hence,
[tex]\[ S = kx + \frac{k(k-1)}{2} \][/tex]
- Since \(S\) must equal 585,
[tex]\[ kx + \frac{k(k-1)}{2} = 585 \][/tex]

4. Solve for \(x\):
- Rearrange the equation to solve for \(x\):
[tex]\[ kx + \frac{k(k-1)}{2} = 585 \][/tex]
[tex]\[ kx = 585 - \frac{k(k-1)}{2} \][/tex]
[tex]\[ x = \frac{585 - \frac{k(k-1)}{2}}{k} \][/tex]

5. Determine feasibility:
- For \(x\) to be a positive integer, \(\frac{585 - \frac{k(k-1)}{2}}{k}\) must be a positive integer.
- We will need to check various values of \(k\) and see when \(x\) is a positive integer.

6. Check maximum value of \(k\):
- Upon examining \(k\) and ensuring that \(x\) remains a positive integer, we find that the maximum value for \(k\) that satisfies this condition is indeed 30.

Therefore, the greatest number of consecutive positive integers whose sum is 585 is:

Option (B) 30.