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        riddles >> medium >> Infinite System of Equations
        
(Message started by: ThudanBlunder on Jan 20^th, 2009, 7:38pm)

Title: Infinite System of Equations
Post by ThudanBlunder on Jan 20^th, 2009, 7:38pm

Solve the following infinite system of equations, where N is an integer and A,B,X,Y are distinct, one-digit integers:

A*N = XY
AB*N = XXY
ABB*N = XXXY
ABBB*N = XXXXY
etc.

Title: Re: Infinite System of Equations
Post by balakrishnan on Jan 21^st, 2009, 1:05am

[hide]The solution set
{A,B,X,Y,N} can be
{3,1,7,5,25}
{8,3,6,4,8}
and
{8,3,9,6,12}[/hide]

Title: Re: Infinite System of Equations
Post by codpro880 on Mar 3^rd, 2009, 6:04pm

They all seem to work for the first few equations....but how did you come up with those answers and how can you prove they'll always work?

Title: Re: Infinite System of Equations
Post by Eigenray on Mar 4^th, 2009, 12:33am

The first equation tells us obviously A*N = 10X + Y. Now suppose
AB...B*N = XX...XY. Then
AB...BB*N = [10*AB...B + B]*N
= 10*(AB...B*N) + B*N
= XX...XY0 + B*N
So to get from one step to the next, it is necessary and sufficient that Y0 + B*N = XY. Since also XY = A*N, we get N*(A-B) = 10*Y. So for each value of Y we can just look at the divisors of 10Y and see which pairs work for N, A-B.

I actually did it differently: divide the n-th line by 10^n-1. Then
A.B...B * N = XX.X...Y,
and taking the limit gives (A + B/9)*N = X*100/9. Combined with A*N=XY this is equivalent to Y0+B*N=XY, but if you do it this way I guess it's less obvious that any solution will work.