Find the smallest natural number $n$

Question

Find the smallest natural number $n$ such that rightmost digit is $6$ and when we deleted that digit $6$ and add it to the left of the number we get $4n$.

Example of the operation:

$123456$ becomes $612345$

$16$ becomes $61$

$66$ becomes $66$

Answer 1

Just start playing around with some numbers. So far, we know that:

...FEDCBA6
       x 4
----------
6...FEDCBA

where we can find each digit A, B, C, ... one by one until we reach a 6. Time to do some elementary school arithmetic: 6 * 4 = 24, so A is 4 and we can carry the 2:

        2
...FEDCB46
       x 4
----------
6...FEDCB4

Likewise, 4 * 4 + 2 = 18, so B = 8 and we carry the 1:

       12
...FEDC846
       x 4
----------
6...FEDC84

Can you continue from here?

Answer 2

If $n$ ends with 6, then $4n$ ends with 4 (because $6\cdot 4 = 24$). So the tail looks like

.....6 * 4 = .....4

Then we know $n$ ends with 46 – and so on:

....46 * 4 = ....84
...846 * 4 = ...384
..3846 * 4 = ..5384
.53846 * 4 = .15384
153846 * 4 = 615384

Done.

An example of a larger number with the same property is 153846153846153846.

Answer 3

$n = 10x + 6$ where $10^{d-1} \le x \le 10^d - 1$.

$4n = 6 \cdot 10^d + x$

$40x + 24 = 6 \cdot 10^d + x$

$39x + 24 = 6 \cdot 10^d$

$13x + 8 = 2 \cdot 10^d$

$2\cdot 10^d \equiv 8 \pmod{13}$

$10^d \equiv 4 \pmod{13}$

 Multiples of 13: 13, 26, 39, 52, 65, 78, 91, 104, 117

\begin{align} 10^2 &\equiv 100 \equiv 9 \pmod{13}\\ 10^3 &\equiv 90 \equiv -1 \pmod{13}\\ 10^4 &\equiv -10 \equiv 3 \pmod{13}\\ 10^5 &\equiv 30 \equiv 4 \pmod{13} \end{align}

The smallest such $d$ is $d=5$.

$13x + 8 = 200000 \implies x = 15384$

So $n = 153846$

Alternate solution

$2 \cdot 10^d = 13x + 8$

$10^d \cdot \dfrac{2}{13} = x + \dfrac{8}{13}$

Note $\dfrac{2}{13} = 0. \overline{153846}$ and $\dfrac{8}{13} = 0. \overline{615384}$

So

$10^5 \cdot \dfrac{2}{13} = 15384 + \dfrac{8}{13}$

So $n = 153846$