# Task 6. (14 points)

Natural numbers $a, b, c, d$, and $e$ are consecutive terms of an arithmetic progression. Find the smallest possible value of the number $c$, if the sum $b+c+d$ is a perfect square, and the sum $a+b+c+d+e$ is a perfect cube.
# Solution.

Since $b+d=2c$, then $3c=n^2$ for some natural number $n$.

Therefore, $n$ is divisible by 3 and $c=3l^2$ for some natural number $l$.

Since $a+b+d+e=4c$, then $5c=m^3$ for some natural number $m$.

Therefore, $m$ is divisible by 5 and $c=5^2 l^3$ for some natural number $l$.

The smallest number that satisfies these conditions is $5^2 3^3=675$.

Answer: 675.

| Content of the criterion | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | ![](https://cdn.mathpix.com/cropped/2024_05_06_c5fbf2b9d026492118aeg-6.jpg?height=69&width=176&top_left_y=1867&top_left_x=1776) |
| Main logical steps of the solution are presented. Some justifications are missing in the solution. The answer is correct. | $\pm$ | 11 |
| Main logical steps of the solution are presented. The answer is incorrect. | $+/ 2$ | 7 |
| The solution contains a certain advancement in the right direction. The answer is correct. |  |  |
| The solution is generally incorrect or incomplete, but contains a certain advancement in the right direction. The answer is incorrect or missing. | 干 | 3 |
| The answer is correct. The solution is missing or incorrect. |  |  |
| The solution does not meet any of the above criteria. | -10 | 0 |