TITLE: Find all natural numbers with the property that...
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: Besides EuYu's, another way to arrive at the answer is to say that the leading digit has to be $1$ or $2$, because if it were $3$ or greater there would be a carry and the product would have more digits than the original number.  Let's try $1$.  The multiplicand then starts with $1$ and the product ends with $1$.  To have the product end with $1$, the multiplicand must end with $6$ as $6 \cdot \frac 72=21$.  Then the product ends with $61$ and the multiplicand ends with $46$.  We keep going until we have a $1$ at the front, giving $$ 153846 \\ \underline{\times \ \ \ \ 3.5} \\ 538461$$  Now we can see that multiplying by $2$ will carry, so will not work and this is the only primitive solution.