Let ^@S^@ be the smallest positive multiple of ^@15^@ that compr | Math Question and Answer

Answer:

$0$

Step by Step Explanation:

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 $5$ .
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 $S$ .
$\implies S = 300338880$

The remainder when

S S

$S$ is divided by

88

$8$

= =

$=$ Remainder of (Last

33

$3$ digits of

S \div 8 S \div 8

$S \div 8$ )

= =

$=$ Remainder of (

880 \div 8 880 \div 8

$880 \div 8$ )

= 0 = 0

$= 0$

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