Find the smallest number such that if its rightmost digit is placed at its left end, the new number so formed is precisely 50% larger than the original number.

There's an standard solution using continued fractions. Assume we are looking for a number abc...z.

Let's write the fraction x=0.abc...zabc...z... Therefore, (x+z)/10= 0.zabc...zabc...zabc...z..., which should be 1.5x.

Equating members, x=z/14. With z=4, we get x=0.285714, which ends in 4 as needed, so the number is 285714.