One Before, One After

Find a five-digit number with this property: writing a $1$ after it gives a number three times as large as writing a $1$ before it.

Solution

Call the number $N$. Writing a $1$ after it appends a digit, making $10N + 1$; writing a $1$ before it puts a $1$ in the hundred-thousands place, making $100000 + N$. The condition is $$10N + 1 = 3\,(100000 + N),$$ so $10N + 1 = 300000 + 3N$, giving $7N = 299999$ and $N = 42857$. And indeed $$428571 = 3 \times 142857.$$ That $142857$ should surface here is no coincidence. It is the repeating block of $\tfrac17 = 0.\overline{142857}$, the best known of the cyclic numbers, and our answer $42857$ is simply $142857$ with its leading $1$ struck off.