IrMO 1992 Problem 7

If $a_1$ is a positive integer, form the sequence $a_1,a_2,a_3,\dots$ by letting $a_2$ be the product of the digits of $a_1$ , etc.. If $a_k$ consists of a single digit, for some $k\ge 1$ , $a_k$ is called a digital root of $a_1$ . It is easy to check that every positive integer has a unique root. $($ For example, if $a_1=24378$ , then $a_2=1344$ , $a_3=48$ , $a_4=32$ , $a_5=6$ , and thus $6$ is the digital root of $24378.)$ Prove that the digital root of a positive integer $n$ equals $1$ if, and only if, all the digits of $n$ equal $1$ .