# Talk:Digital root/Multiplicative digital root

The second task "tabulate MP versus the first five numbers having that MP" is inconsistent with the sample output. Assuming the output values are correct, the task should say "tabulate MDR versus the first five numbers having that MDR", and the "MD" column in the output should be "MDR". --Globules 06:00:08, 20 April 2014 (UTC)

I've fixed that. Values with an MP of 9 seem to be rather large (I stopped looking at 20000000). --Rdm (talk) 08:39, 20 April 2014 (UTC)

I've promoted this to a task. It's got a clear description, and it's got more than 4 implementations in different languages. –Donal Fellows (talk) 15:58, 27 April 2014 (UTC)

## The product of decimal digits must be a humble numbers ( 2^a*3^b*5^c*7^d )

Decimal digits 2..9 are humble numbers
1 does not change anything. 0 stops. <lang pascal>program MultRoot; {\$IFDEF FPC}

``` {\$MODE DELPHI}{\$OPTIMIZATION ON,ALL}
```

{\$ENDIF} {\$IFDEF WINDOWS}

``` {\$APPTYPE CONSOLE}
```

{\$ENDIF} uses

``` sysutils;
```

const

``` //mul digit of 277777788888899 = 4996238671872
lnMax = ln(4996238671873);//ln(High(Uint64));
```

type

``` tnm = record
nmNum : Uint64;
nmLnNum : double;
nmPots: array[0..3] of byte;
nmMulRoot,
nmMulPers : Int16;
end;
tHumble = array[0..4679{15540}] of tnm;
```

var

``` Humble : tHumble;
idx: Uint32;
```

Procedure QuickSort ( Left, Right : LongInt ); Var

``` i, j : LongInt;
pivot : Uint64;
tmp : tnm;
```

Begin

``` i:=Left;
j:=Right;
pivot := Humble[(Left + Right) shr 1].nmNum;
Repeat
While pivot > Humble[i].nmNum Do inc(i);
While pivot < Humble[j].nmNum Do dec(j);
```
```   If i<=j Then Begin
tmp:=Humble[i];
Humble[i]:=Humble[j];
Humble[j]:=tmp;
dec(j);
inc(i);
End;
Until i>j;
If Left<j Then QuickSort(Left,j);
If i<Right Then QuickSort(i,Right);
```

End;

function GetMulDigits(n:Uint64):UInt64; //inserting only numbers without any '0' var

``` i,q :Uint64;
```

begin

``` i := 1;
repeat
q := n div 10;
i := (n-10*q)*i;
n := q;
until (i= 0) OR (n= 0);
GetMulDigits := i;
```

end;

procedure Insert(prime,NumIdx:Uint32); var

``` lnPot,
lnPotSum:double;
potNum : Uint64;
i,j,pot :Uint32;
```

begin

``` i := idx+1;
pot := 0;
potNum := 1;
lnPot := ln(prime);
lnPotSum := 0.0;
repeat
inc(pot);
potNum := potNum*prime;
lnPotSum := pot*lnPot;
if lnPotSum>lnMax then
BREAK;
for j := 0 to idx do
begin
//ends in '0' 2^x*5*y //x,y > 0 will stay 0
if (numIdx = 2) AND (Humble[j].nmPots[0]<> 0) then  continue;
Humble[i] := Humble[j];
with Humble[i] do
begin
if (Potnum>0) AND (nmLnNum+lnPotSum < lnMax) then
begin
nmLnNum := nmLnNum+lnPotSum;
nmNum := nmNum*potNum;
nmPots[NumIdx] := pot;
nmMulRoot := -1;
nmMulPers := 0;
inc(i);
end;
end;
end;
until false;
idx := i-1;
```
``` writeln('insert powers of ',prime,' new count ',idx);
```

end;

procedure OutHumble(h:tnm); var

``` s : string[23];
n,last : UInt64;
i,p : Uint32;
ch: char;
```

begin

``` with h do
begin
write(h.nmMulPers:3,' : ');
n := nmNum;
For i := 0 to 3 do write(nmPots[i]:3);
//creating smallest number which digits multiply to n
setlength(s,23);
//extract '9'
s[1] := ' ';
p:= 2;
while nmPots[1]>1 do
begin
s[p] :=('9');inc(p);
nmPots[1] := nmPots[1]-2;
end;
//'8'
while nmPots[0]>2 do
begin
s[p] :=('8');inc(p);
nmPots[0] := nmPots[0]-3;
end;
//'7'
while nmPots[3]>0 do
begin
s[p] :=('7');inc(p);
nmPots[3] := nmPots[3]-1;
end;
//'6'
while (nmPots[0]>0) AND (nmPots[1]>0) do
begin
s[p] :=('6');inc(p);
nmPots[0] := nmPots[0]-1;
nmPots[1] := nmPots[1]-1;
end;
//'5'
while (nmPots[2]>0)do
begin
s[p] :=('5');inc(p);
nmPots[2] := nmPots[2]-1;
end;
//'4'
while (nmPots[0]>1)do
begin
s[p] :=('4');inc(p);
nmPots[0] := nmPots[0]-2;
end;
//'3'
if (nmPots[1]>0) then
begin
s[p] :=('3');inc(p);
end;
//'2'
if nmPots[0]>0 then
begin
s[p] :=('2');inc(p);
end;
i := 2;
p := p-1;
setlength(s,p);
//swap digits
```

while i

= 10 then write(s,'->',n) else write(' ',n); last := n; // n := GetMulDigits(n); if last <> n then begin repeat write('->',n); last := n; n := GetMulDigits(n); until last=n; end; writeln; end; end; var n,last : Uint64; i,j : Uint32; begin Humble[0].nmNum :=1; Insert(2,0); Insert(3,1); Insert(5,2); Insert(7,3); //remove numbers with one '0' digit j:= 0; For i := 0 to Idx do begin if GetMulDigits(Humble[i].nmNum) <> 0 then Begin Humble[j] := Humble[i]; inc(j); end; end; idx := j-1; writeln('remove numbers with "0" digit.Remaining ',idx); QuickSort(0,idx); For i := 0 to Idx do begin j :=0; n := Humble[i].nmNum; last := n; n := GetMulDigits(n); if last <> n then begin j := 1; repeat inc(j); last := n; n := GetMulDigits(n); until last=n; end; Humble[i].nmMulRoot:= n; Humble[i].nmMulPers:= j; end; For i := 0 to idx do OutHumble(Humble[i]); {\$IFDEF WINDOWS} write(' done. Press <ENTER>');readln; {\$ENDIF} end. Whats special about 277777788888899 277,777,788,888,899 </lang>

@TIO.RUN:
```//Real time: 0.134 s User time: 0.094 s
insert powers of 2 new count 42
insert powers of 3 new count 595
insert powers of 5 new count 833
insert powers of 7 new count 4679
remove numbers with "0" digit.Remaining 2096
mulpersistance : pot 2,3,5,7
0 :   0  0  0  0      1
0 :   1  0  0  0      2
0 :   0  1  0  0      3
0 :   2  0  0  0      4
0 :   0  0  1  0      5
0 :   1  1  0  0      6
0 :   0  0  0  1      7
0 :   3  0  0  0      8
0 :   0  2  0  0      9
2 :   2  1  0  0 26->12->2
2 :   1  0  0  1 27->14->4
2 :   0  1  1  0 35->15->5
2 :   4  0  0  0 28->16->6
2 :   1  2  0  0 29->18->8
2 :   0  1  0  1 37->21->2
2 :   3  1  0  0 38->24->8

....  267777777899999->smallest number with mul dgt of -> humble 2^5*3^11*5^0*7^7  =4668421498272

6 :   5 11  0  7 267777777899999->4668421498272->74317824->37632->756->210->0
3 :   6 21  0  1 37889999999999->4686238234944->191102976->0
3 :   0 13  2  6 355777777999999->4689262665675->1567641600->0
3 :  31  7  0  0 68888888888999->4696546738176->1097349120->0
3 :  11  9  0  6 267777778889999->4742523426816->15482880->0
3 :   0  3  4 10 3555577777777779->4766769826875->10241925120->0
3 :   2 20  0  3 47779999999999->4783868198172->260112384->0
3 :  27  6  0  2 77888888888999->4794391461888->334430208->0
3 :   0  1  9  7 35555555557777777->4825447265625->129024000->0
3 :  23  5  0  4 267777888888899->4894274617344->130056192->0
4 :  13  6  0  7 277777778888999->4918172442624->6193152->1620->0
11 :  19  4  0  6 277777788888899->4996238671872->438939648->4478976->338688->27648->2688->768->336->54->20->0```