[Guest]

Determine all possible octuplet(s) (P, Q, R, S, T, U, V, W) of positive integers, with P ≤ Q ≤ R ≤ S, and T ≤ U ≤ V ≤ W, and P < T satisfying: P+Q+R+S = T*U*V*W and, P*Q*R*S = T+U+V+W.

[Guest]

For given conditions. If P>2 , P+Q+R+S < P*Q*R*S

Since T> P, T+U+V+W < T*U*V*W

If P*Q*R*S=T+U+V+W then P+Q+R+S < T*U*V*W.

For P=1, I could not find any solution.

Narendra

[Guest]

For given conditions. If P>2 , P+Q+R+S < P*Q*R*S

Since T> P, T+U+V+W < T*U*V*W

If P*Q*R*S=T+U+V+W then P+Q+R+S < T*U*V*W.

For P=1, I could not find any solution.

Narendra

You are right, NS.

Looking back at the original post, I have detected a typographical lacuna in the second line - where, in place of the condition P < T, it should have been, P <= T.

In view of the maximum time limit of 10 minutes for any given submitter to correct his original post, I visualize no succinct avenue to incorporate the said amendment in the OP.

Edited December 26, 2009 by K Sengupta

[Guest]

Ok I could not find a solution for P < T either but

First let me say that I used a brute force approch to this so I may not have all of them as I limited the numbers to <=10. I know thats a small number but the number of permutations was already 100000000 and using numbers <=11 more than doubles the permutations. In narendrasony's post it was concluded that there are no solutions for P < T so it seems that you could set P = T of which I found 3.

P  Q  R  S  T  U  V  W


1  1  1  9  1  1  3  4  Both sides + and * to 12


1  1  2  4  1  1  2  4  Both sides + and * to 8


1  1  3  4  1  1  1  9  Both sides + and * to 12 and it is the same as the first with PQRS switched with TUVW

Edit: spelling errors

Edited December 26, 2009 by stewdeker

[Guest]

If P=T then one of the quadrapulets (say P,Q,R,S) should satisfy : P+Q+R+S >= PQRS. This can be true iff P=1 and Q=1 (verify for Q>1 this is not true). So the possible solutions are (P,Q,R,S), (T,U,V,W):

a) (1,1,2,4), (1,1,2,4)

b) (1,1,1,9), (1,1,3,4)

c) (1,1,1,11), (1,1,2,7)

First two are already mentioned by Stewdeker.