Re: Diophantic equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg32771] Re: [mg32746] Diophantic equations*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Sat, 9 Feb 2002 05:11:42 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Here is how to do it in your case. You can use this method to solve more general linear cases. First eliminate one variable from the equations: In[1]:= eqs={a+b+10 c==100,5 a+2 b+c==100} In[2]:= Eliminate[eqs,{c}] Out[2]= 900-19 b==49 a Clearly we only need to find all integer solutions of the equation 49 a + 19 b == 900, since c will then be given by c=100-a-2b. The next step is to find just a single integer solution. The general way to do this is to use the Euclidean algorithm. In[3]:= ExtendedGCD[49,19] Out[3]= {1,{7,-18}} This tells us that: In[4]:= 7*49-18*19 Out[4]= 1 From this we see at once that a solution will be given by: In[7]:= a=900*7 Out[7]= 6300 In[8]:= b=900*-18 Out[8]= -16200 indeed In[9]:= 49 a+19 b Out[9]= 900 Now that we have one solution (a,b) all the other solutions are given by (a+m 19,b-m 49) = (6300+ m*19,-16200-m*49) where m is any integer. Indeed: In[11]:= {6300+m*19,-16200-m*49}.{49,19}//Simplify Out[11]= 900 On Friday, February 8, 2002, at 05:49 PM, Max Ulbrich wrote: > Hi, > > I try to solve equations where only integers are allowed, like > > a + b + 10 c = 100 > 5 a + 2 b + c = 100 > > a, b, c should be integers and positve. > > How can I do this? > > Max > > > > > Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/