Find all natural numbers with the property that... Find all natural numbers with the property that when the first digit is moved to the end, the resulting number is $3.5$ times the original number.

Suppose that our original number is $k+1$ digits long. Let $a$ denote the first digit of the number. We represent the number as
$$10^ka + b$$
where $b < 10^k$ is the remaining portion of our number. The condition is thus expressed as
$$3.5(10^ka + b) = 10b + a$$
Clearing fractions and rearranging, we end up with
$$(7\cdot 10^k - 2)a = 13b$$
Clearly $13\nmid a$ and we must therefore have
$$7\cdot 10^k - 2 \equiv 0 \pmod{13} \implies 10^k \equiv 4 \pmod{13}$$
We find $k = 5$ as the smallest solution to the congruence. The order of $10$ modulo $13$ is $6$, so we have $k\equiv 5\pmod6$. 
Getting back to our former equation, we note that 
$$a \ge 2 \implies \frac{14}{13}10^k - 2 > 10^k$$
which will contradict the restrictions on $b$. We then have $a = 1$. Therefore the numbers are of the form
$$10^k + \frac{7\cdot 10^k - 2}{13} = \frac{2\cdot 10^{k+1} - 2}{13}$$
where $k \equiv 5 \pmod6$. The order of $10$ modulo $13$ is $6$ so the fraction $\frac{1}{13}$ repeats with period $6$. Denote the repeating block of $13^{-1}$ as $r = 076923$. Our number is of the form
$$2\times \frac{10^{6n}- 1}{13} = 2\times\underbrace{rr\cdots r}_{n \text{times}}$$
where the latter expression denotes the number formed by concatenating $r$ with itself $n$ times for some natural number $n$. Putting it all together, our number is of the form $153846$ appended with itself an arbitrary number of times,
$$\left\{153846,\ 153846153846,\ 153846153846153846,\ \cdots\right\}$$
