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.

Let a=A-1    b=B-2....     e=E-5

Let P= abcde  and S=A+B+...E

Considering the range  and the ascending order constraints, and starting with (1,1,1,1,p),then (1,1,1m,p)  etc one quickly arrives to 28 and 40 as  the only solutions of P=S.

Now permuting (1,1,2,2, 7)  - 6 combinations with 7 as  last number and permuting 1,1,1,2, 20 - 4 combinations with 20 as  last number -yields the 1O  sets of (abcde) from which the corresponding sets of (ABCDE) are derived.

Posted by Ady TZIDON on 2010-05-24 14:09:34