Determine all possible quintuplet(s) (A,B,C,D,E) of positive integers, with A ≤ B ≤ C ≤ D ≤ E, that satisfy this equation:
(A-1)*(B-2)*(C-3)*(D-4)*(E-5) = A+B+C+D+E
Prove that these are the only quintuplet(s) that exist.
FOR a = 1 TO 25
FOR b = a TO 26
FOR c = b TO 27
FOR d = c TO 28
FOR e = d TO 29
lhs = (a - 1) * (b - 2) * (c - 3) * (d - 4) * (e - 5)
rhs = a + b + c + d + e
IF lhs = rhs THEN PRINT a; b; c; d; e, lhs; rhs
NEXT
NEXT
NEXT
NEXT
NEXT
produces
A B C D E LHS RHS
2 3 4 6 25 40 40
2 3 5 5 25 40 40
2 3 5 6 12 28 28
2 4 4 5 25 40 40
2 4 4 6 12 28 28
2 4 5 5 12 28 28
3 3 4 5 25 40 40
3 3 4 6 12 28 28
3 3 5 5 12 28 28
3 4 4 5 12 28 28
In (A-1)*(B-2)*(C-3)*(D-4)*(E-5) = A+B+C+D+E
The RHS <= 5*E.
If D > 20 and therefore E > 20, LHS > 16*(E-5), so
LHS > 16*E - 80 = 5*E + 11*E - 80 > 5*E + 220 - 80 = 5*E + 140 > RHS
So the cases covered by the program exhaust the range over which the LHS could equal the RHS.
The limits and argument could be made tighter, but the numbers chosen are sufficient for the proof.
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Posted by Charlie
on 2010-05-24 13:44:42 |
computer solution + proof
