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Re: Solving equations and inequalities with Reduce - how?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87723] Re: [mg87655] Solving equations and inequalities with Reduce - how?
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Wed, 16 Apr 2008 05:02:59 -0400 (EDT)
  • References: <200804150950.FAA24909@smc.vnet.net>

Marc Heusser wrote:
> I tried to solve equations with Reduce and somehow did not quite 
> formulate it right, so Reduce complains with 
> "Reduce::ivar: 1 is not a valid variable".
> 
> This is what I tried:
> 
> Wanted: A six digit number satisfying the following conditions:
> The first digit is not zero.
> If you take the first two digits and move them to the end of the number, 
> the resulting number must be twice the original number.
> 
> In[23]:=Reduce[200000a +20000 b +2000 c+200 d+20 e  +       2 f 
> \[Equal]100000c +10000 d +1000 e+100 f+10 a  + b ,  {a,b,c,d,e,f}, 
> Modulus\[Rule]9]
> 
> I did solve the problem through exhaustive search:
> 
> In[14]:=Timing[Select[Range[10^6],    FractionalPart[#/10000] 1000000 + 
> IntegerPart[#/10000]\[Equal] 2  #&]]
> Out[14]={34.2848 Second,{142857,285714,428571}}
> 
> but would like to understand how to use Reduce (or another function) to 
> solve such a set of equations.
> 
> TIA
> 
> Marc
> 

I realize this has been answered by now, but you might want to have a 
look at problem 1 in the notebook at

http://library.wolfram.com/infocenter/Conferences/6530/

Among other things, I'll note that you really need to enforce "one 
digitedness" by inequalities, not with a Modulus setting. Along the 
lines of that notebook, here is some code for your particular problem.

In[26]:= Reduce[{2*n==100*m+k, n==m+10^4*k, Element[{n,m,k},Integers],
     10^4>m>=10^3, 0<=k<=99, n>=10^5}, {n,m,k}]

Out[26]= (n == 142857 && m == 2857 && k == 14) ||
    (n == 285714 && m == 5714 && k == 28) ||
    (n == 428571 && m == 8571 && k == 42)

Daniel Lichtblau
Wolfram Research


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