OMM 2014 Problem 2

A positive integer $a$ is said to reduce to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$ . For example, 2015 reduces to $\frac{2015}{5} = 403$ . Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.