Primary X-Subfactor Series

Let $n$ be any positive integer. A factor of $n$ is any number that divides evenly into $n$, without leaving a remainder. For example, $13$ is a factor of $52$, since $52/13 = 4$. A subsequence of $n$ is a number without a leading zero that can be obtained from $n$ by discarding one or more of its digits. For example, $2$, $13$, $801$, $882$, and $1324$ are all subsequences of $8013824$, but $214$ is not (you can’t rearrange digits), $8334$ is not (you can’t have more occurrences of a digit than appear in the original number), $8013824$ is not (you must discard at least one digit), and $01$ is not (you can’t have a leading zero). A subfactor of $n$ is an integer greater than $1$ that is both a factor and a subsequence of $n$. $8013824$ has subfactors $8$, $13$, and $14$. Some numbers do not have a subfactor; for example, $6341$ is not divisible by $6$, $3$, $4$, $63$, $64$, $61$, $34$, $31$, $41$, $634$, $631$, $641$, or $341$.

An x-subfactor series of $n$ is a decreasing series of integers $n_1, \ldots , n_ k$, in which (1) $n = n_1$, (2) $k \ge 1$, (3) for all $1 \le i < k$, $n_{i+1}$ is obtained from $n_ i$ by first discarding the digits of a subfactor of $n_ i$, and then discarding leading zeros, if any, and (4) $n_ k$ has no subfactor. The term “x-subfactor” is meant to suggest that a subfactor gets x’ed, or discarded, as you go from one number to the next. For example, $2004$ has two distinct x-subfactor series, the second of which can be obtained in two distinct ways. The highlighted digits show the subfactor that was removed to produce the next number in the series.

2004 4
2004 200 0
2004 200 0

The primary x-subfactor series has maximal length (the largest $k$ possible, using the notation above). If there are two or more maximal-length series, then the one with the smallest second number is primary; if all maximal-length series have the same first and second numbers, then the one with the smallest third number is primary; and so on. Every positive integer has a unique primary x-subfactor series, although it may be possible to obtain it in more than one way, as is the case with $2004$.


The input consists of at most $25$ positive integers, each less than one billion, without leading zeroes, and on a line by itself. Following is a line containing only “0” that signals the end of the input.


For each positive integer, output its primary x-subfactor series using the exact format shown in the examples below.

Sample Input 1 Sample Output 1
123456789 12345678 1245678 124568 12456 1245 124 12 1
2004 200 0
8013824 13824 1324 132 12 1
CPU Time limit 1 second
Memory limit 1024 MB
Difficulty 5.3medium
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License Creative Commons License (cc by-sa)

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