Number Theory Question by Mithil Shah # 2

Let

$x=10a+6$

After the operation we get

$6\cdot 10^m+a$

Where $m$ is the number of digits of $a$ . It is given that

$6\cdot 10^m+a=4(10a+6)$

Solving for $a$

$a=\dfrac{2(10^m-4)}{13}$

Checking powers of $10$ modulo $13$

$\begin{cases} 10^1\equiv 10\mod 13\\ 10^2\equiv 9\mod 13\\ 10^3\equiv 12\mod 13\\ 10^4\equiv 3\mod 13\\ 10^5\equiv 4\mod 13 \end{cases}$

$\Longrightarrow m=5$ is the smallest possible, which gives

$a=\dfrac{2(10^5-4)}{13}=15384$

Hence

$x=15384\cdot 10+6=\boxed{153846}$

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$15384\color{#D61F06}6\color{#333333}\cdot 4=\color{#D61F06}6\color{#333333}15384$