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My favorite numbers VI (Posted on 2010-11-21) Difficulty: 4 of 5
Determine all possible sextuplet(s) (A, B, C, D, E, F), with B < C, E < F and, A < D, that satisfy this system of equations:

A/(B*C) = D - E - F, and:

D/(E*F) = A - B - C

Prove that these are the only sextuplet(s) that exist.

No Solution Yet Submitted by K Sengupta    
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possible approach Comment 6 of 6 |
(In reply to re: One Solution (spoiler) by Steve Herman)

Try these substitutions:

n1=d-e-f;a=n1*bc
n2=a-b-c:d=n2*ef
n1=(n2*ef)-e-f;n1+e+f=(n2*ef) :
n2=(n1*bc)-b-c;n2+b+c=(n1*bc)

It is then clear that we are looking for solutions the smaller of which is bounded by the equation xyz=x+y+z for which the largest solution is xyz=6, a perfect number. So at least one of (a,b,c) is 1 and at least one of (n2,e,f) is 1.The pair {b=1, n2=1} is forced by the terms of the problem since a,c>b and a<d, and b cannot = 2, for then a = d contrary to what was stipulated.

Now ef=d,a=n1*c; but(a-c-1) = 1;so (n1*c=c+2) with {c=1,n1=3} and {c=2,n1=2} but c>b, so c=2, n1 = 2, a = 4.In that case, 2+e+f=(ef), where one of (e,f) is 2 and the other 4, but e<f, so e=2, f=4; d = ef = 8.


  Posted by broll on 2010-11-23 03:09:48
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