user
The multiplicative digital root (MDR) and multiplicative persistence (MP) of a number, 

n
{\displaystyle {\displaystyle n}}
, is calculated rather like the Digital root except digits are multiplied instead of being added:

1. Set 

   m
   {\displaystyle {\displaystyle m}}
    to 

   n
   {\displaystyle {\displaystyle n}}
    and 

   i
   {\displaystyle {\displaystyle i}}
    to 

   0
   {\displaystyle {\displaystyle 0}}
   .
2. While 

   m
   {\displaystyle {\displaystyle m}}
    has more than one digit:
   * Find a replacement 

     m
     {\displaystyle {\displaystyle m}}
      as the multiplication of the digits of the current value of 

     m
     {\displaystyle {\displaystyle m}}
     .
   * Increment 

     i
     {\displaystyle {\displaystyle i}}
     .
3. Return 

   i
   {\displaystyle {\displaystyle i}}
    (= MP) and 

   m
   {\displaystyle {\displaystyle m}}
    (= MDR)

Task

* Tabulate the MP and MDR of the numbers 123321, 7739, 893, 899998
* Tabulate MDR versus the first five numbers having that MDR, something like:

```
MDR: [n0..n4]
===  ========
  0: [0, 10, 20, 25, 30]
  1: [1, 11, 111, 1111, 11111]
  2: [2, 12, 21, 26, 34]
  3: [3, 13, 31, 113, 131]
  4: [4, 14, 22, 27, 39]
  5: [5, 15, 35, 51, 53]
  6: [6, 16, 23, 28, 32]
  7: [7, 17, 71, 117, 171]
  8: [8, 18, 24, 29, 36]
  9: [9, 19, 33, 91, 119]

```

Show all output on this page.

Similar

The Product of decimal digits of n page was redirected here, and had the following description

```
Find the product of the decimal digits of a positive integer   n,   where n <= 100

```

The three existing entries for Phix, REXX, and Ring have been moved here, under ===Similar=== headings, feel free to match or ignore them.

References

* Multiplicative Digital Root on Wolfram Mathworld.
* Multiplicative digital root on The On-Line Encyclopedia of Integer Sequences.
* What's special about 277777788888899? - Numberphile video
assistant
(formerly Perl 6)

```
sub multiplicative-digital-root(Int $n) {
    return .elems - 1, .[.end]
        given cache($n, {[*] .comb} ... *.chars == 1)
}

for 123321, 7739, 893, 899998 {
    say "$_: ", .&multiplicative-digital-root;
}

for ^10 -> $d {
    say "$d : ", .[^5]
        given (1..*).grep: *.&multiplicative-digital-root[1] == $d;
}

```

Output:

```
123321: 3 8
7739: 3 8
893: 3 2
899998: 2 0
0 : 10 20 25 30 40
1 : 1 11 111 1111 11111
2 : 2 12 21 26 34
3 : 3 13 31 113 131
4 : 4 14 22 27 39
5 : 5 15 35 51 53
6 : 6 16 23 28 32
7 : 7 17 71 117 171
8 : 8 18 24 29 36
9 : 9 19 33 91 119
```