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Simultaneous Settlement II (Posted on 2013-09-26) Difficulty: 3 of 5
Determine the total number of nonnegative integer solutions to this system of simultaneous equations:

(A) P*Q*R*S*T = 7X*11Y*13Z*79W, and:
(B) X+Y+Z+W = 44

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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analytical solution | Comment 1 of 3

first, consider any integer solution to the equation
x+y+z+w=44
where each of x,y,z,w is nonnegative.
for each of these we get a unique number
7^x*11^y*13^z*79^w because each of 7,11,13, and 79 are prime.
Now for each of these, we need to determine how many ways we can distribute each of these prime factors amoung the five factors p,q,r,s and t.  So for example, if we have x=2, then we have two factors of 7 to distribute among the 5 factors.  So what we are really asking is how many nonnegative integer solutions to a+b+c+d+e=2 as this gives p=7^a,q=7^b and so forth.
Now this number can be built up from simpler problems as such:

first consider how solutions there are to a=x, there is only 1.
so let f1(x)=1
now consider number of solutions to a+b=x, this is simply
f2(x)=sum(f1(x-a),a=0 to x)
similarly we can build the other formulas from the previous one
f3(x)=sum(f2(x-a),a=0 to x)
f4(x)=sum(f3(x-a),a=0 to x)
f5(x)=sum(f4(x-a),a=0 to x)
giving us
f5(x)=(x+1)(x+4)(x^2+5x+6)/24

Now to get the number of solutions to the system of equations we need only take the sum of
f5(x)*f5(y)*f5(z)*f5(44-x-y-z) with
x=0 to 44
y=0 to 44-x
z=0 to 44-x-y

Putting this nasty nested summation into my CAS on my phone gives the big number
6,131,164,307,378,475

my next step is to obtain a closed form equation for the number of solutions with x+y+z+w=n.  However this is quite beyond the ability of my phones CAS and will have to wait for me to get home to my copy of Mathematica :-).


  Posted by Daniel on 2013-09-26 11:28:03
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