### The letters $a, b$ and $c$ stand for non-zero digits. The integer $abc$ is a multiple of $3$ the integer $cbabc$ is a multiple of $15,$ and the integer $abcba$ is a multiple of $8.$ What is the value of the integer $cba?$

$576$

Step by Step Explanation:
1. We know that a number is divisible by $8$ if it's last $3$ digits are divisible by $8.$
Given, $abcba$ is a multiple of $8.$
Therefore $cba$ is a multiple of $8.$
2. Also, $abc$ is given to be a multiple of $3.$
Since the sum of the digits of $abc$ and $cba$ are the same, $cba$ is also a multiple of $3.$
Therefore, $cba$ is a multiple of $24.$
3. We are given that $cbabc$ is a multiple of $15$ and $c \ne 0$ (given).
$\implies c = 5$
Now, $cbabc$ is a multiple of $15$ therefore $cbabc$ is a multiple of $3.$
$\implies$ sum of digits of $cbabc$ is a multiple of $3.$
Also, $a + b + c$ is a multiple of $3,$ therefore, $c + b$ is a multiple of $3.$
4. The three-digit multiples of $24$ starting with $5 ,$ which are the possible values of $cba$ are $504, 528, 552,$ and $576.$
Out of the above possible values of $cba,$ only $576$ has $c + b$ as a multiple of $3.$
5. Hence, the value of the integer $cba$ is $576.$