Probability of sum to be divisible by 7 6 fair dice are thrown simultaneously. What is the probability that the sum of the numbers appeared on dice is divisible by 7 ?

We could approach this via markov chains and matrices.
$A=\begin{bmatrix}0&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}\\
\frac{1}{6}&0&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}\\
\frac{1}{6}&\frac{1}{6}&0&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}\\
\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&0&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}\\
\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&0&\frac{1}{6}&\frac{1}{6}\\
\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&0&\frac{1}{6}\\
\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&0\end{bmatrix}$
Each entry $A_{i,j}$ corresponds to the probability of moving from the $j^{th}$ state to the $i^{th}$ state, where each state corresponds to the remainder of the sum of the rolls modulo $7$.
With initial state vector $v=\begin{bmatrix}1\\0\\0\\0\\0\\0\\0\end{bmatrix}$ we have:
The $i^{th}$ entry of $A^nv$ denotes the probability that after $n$ dice are tossed that the total is $i-1$ more than a multiple of $7$.  In particular, the first entry of $A^6v$ will be the probability that the sum of six dice is a multiple of seven.

As @ThomasAndrews points out, the tedium of this process can be cut down dramatically by noting that we could just as easily have described it as a two-state markov chain.  Either it is or it isn't a multiple of seven.
To see how this looks, we have:
$A=\begin{bmatrix}0&\frac{1}{6}\\1&\frac{5}{6}\end{bmatrix}$ with initial state $v=\begin{bmatrix}1\\0\end{bmatrix}$
We have then the first entry of $A^n v$ will correspond to the probability that after $n$ rolls of the dice the sum is a multiple of $7$, and the second entry corresponds to not being a multiple of $7$.  The original matrix could give us additional information such as probabilities of having specifically remainder $3$ or $4$ modulo $7$ which this simplified matrix could not, but such information is irrelevant to the stated problem.
