The letters ^@ a, b ^@ and ^@ c ^@ stand for non-zero digits. Th | Math Question and Answer

Answer:

$576$

Step by Step Explanation:

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 $8.$
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 $24.$
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 $3.$
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 $3.$
Hence, the value of the integer $cba$ is $576.$

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