### Let $S$ be the smallest positive multiple of $15$ that comprises exactly $3k$ digits with k ‘0’s, k ‘3’s and k ‘8’s. Find the remainder when $S$ is divided by $8$.

$0$

Step by Step Explanation:
1. If a number is a multiple of $15$, it is a multiple of $3$ and $5$ both.
We are given that $S$ is the smallest positive multiple of $15$ which comprises exactly $3k$ digits. Also, $S$ has $k$ ‘$0$’$s$, $k$‘$3$’$s,$ and $k$‘$8$’$s.$
Observe that $S$ must end with $0$ as it is a multiple of $5$.
2. The sum of all the digits of $S = k \times 0 + k \times 3 + k \times 8 = 3k + 8k = 11k$
Since $S$ is a multiple of $3$, the sum of all its digits must be a multiple of $3$.
The smallest value of $k$ such that $11k$ is a multiple of $3$ is $3$. Therefore, there are $3$‘$0$’$s$, $3$‘$3$’$s,$ and $3$‘$8$’$s$ in $S$.
$\implies S = 300338880$
3.  The remainder when $S$ is divided by $8$ $=$ Remainder of (Last $3$ digits of $S \div 8$) $=$ Remainder of ($880 \div 8$) $= 0$
4. Hence, the remainder when $S$ is divided by $8$ is $0$.