Problems & Puzzles: Puzzles

Puzzle 806. Extended n-Parasitic numbers

Jan van Delden sent the following nice puzzle:

An n-parasitic number (in base 10) is a positive natural number which can be multiplied by n by moving the rightmost digit of its decimal representation to the front. Here n is itself a single-digit positive natural number.
[Definition from eng.wikipedia.org]
 
157894736842105263 x 2 =
315789473684210526
 
179487 x 4 =
717948
 
Letís extend the definition of parasitic number in the following way:
 
An extended n-parasitic number (in base 10) is a positive natural number which can be multiplied by n by moving the rightmost digits of its decimal representation to the front. Here n is itself a single-digit positive natural number.
 
103678929765886287625418060200668896321070234113712374581939799331 x 3=
311036789297658862876254180602006688963210702341137123745819397993
 
The extended n-parasitic number is therefore a number m such that m=a.b (the concatenation of a and b) and  n x a.b=b.a. In the given examples b is a prime number.
 
It is straightforward to show that the first digit of b must be at least n. In the questions we assume n>1 (to keep it interesting).
 
Q1:    Show that there is at least one extended n-parasitic number m for which a<b. Why are these solutions rare?
 
Q2:    Are there extended n-parasitic numbers where the number of digits of a is smaller than the number of digits of b?
 
Q3:    Is m always composite?

Q4:    Show that an extended n-parasitic number m=a.b exists for every n and for every b prime (first digit of b >= n) or give a counterexample.
 
Q5:    Find the smallest extended n-parasitic number m, having the last digit of b in {1,3,5,7,9} and b prime.


Contribution came from Emmanuel Vantieghem

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Emmanuel wrote:

Q1, Q2 and Q4.
Let  m = a.b  be the concatenation of the k-digit  a  with the  s-digit  b  and suppose  m  is extended n-parasitic.
Then we have : n*(a*10^s + b) = b*10^k + a.
Thus : a*(n*10^s - 1) = b*(10^k - n).     (***)
Set  D = n*10^s - 1  and  T = 10^k - n.
It is easy to find solutions of  (***)  for given  n  and  s  by choosing  k  such that  T  is divisible by  D.
Indeed, T == 0 (mod D)  means :
10^k == n == 10^(-s) (mod D) or :
10^(k+s) == 1 (mod D).
That means that  k+s  must be equal to the multiplicative order  u  of  10  mod D.
Since  D  and  10  have no divisor in common, this  u  certainly exists..
(I used Mathematica to compute every  u).
This procedure gives lots of solutions (but not all) in which  k  will be bigger than  s  by choosing any  b  with s digits and leading digit >= n.
To find solutions with given  k and  n  we can try to make  D  divisible by  T.  But this is not allways evident since  10  and  T  may have a divisor in common.
Therefore these solutions are less abundant (but not at all 'rare' !).  Here are some of them :
3*142857 = 428571
9*109890 = 989010
7*118279569892473 = 827956989247311
Remark : there are also solutions in which  n  and  10 have a common divisor. Here is one :
5*157894736842105263 = 789473684210526315

 
Q3
All n-parasitic numbers  m  are divisible by 3.
Proof :
Let  z  be the sum of the digits of  m.
Since  n*m  and  m  are composed of the same digits, we have modulo 9 :
n*z == z.  If  m  would be  prime, z  has no divisors in common with  9, hence  n == 1 mod 9.
Since the case  n = 1  is out of consideration,we have QED.

 
Q5.
The smallest (extended) n-parasitic number is  128205 (n = 4)

...

I computed the sequence of all extended n-parasitic numbers.
It was not very difficult to find all the terms < 1000000 of that sequence :
102564, 109890, 128205, 142857, 153846, 179487, 190476, 205128, 230769, 238095, 285714, 307692, 428571.
But, to find the next terms is not at all easy.  I think the next is  1012658227848  but I'm not a 100% sure.. And what's the second next ?
[What is the n value for each of your members sent?]... Some of them have more than one  n.  The sets of values n are respectively :
{4}, {9}, {4}, {2, 3, 4, 5, 6}, {3, 4}, {4}, {4}, {4}, {3, 4}, {4}, {2, 3}, {3}, {2}... I think  102564102564  is a better candidate to be the next term (and not the 13-digit 1012658227848  I mentioned in the previous mails).

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