Each of P, Q, R, S and T
are positive integers
with P < Q < R < S < T
. Determine the maximum value
of the following expression.
[P, Q] -1 + [Q, R] -1 + [R, S] -1 + [S, T] -1
: [x, y] represents the LCM
of x and y.
(In reply to a stab
I can show that (1,2,4,8,16) is maximum.
with x<y for 1/[x,y] to be maximized we want [x,y] to be minimized and obviously [x,y] can't be smaller than y and this is only true if y=x*k for some k
thus we want
for intergers k1,k2,k3,k4>1
thus we want to minimize
1/[P,Q] + 1/[Q,R] + 1/[R,S] + 1/[S,T]
substituting and and extracting the (1/P) common factor we get
(1/P)*(1/k1 + 1/(k1k2) + 1/(k1k2k3) + 1/(k1k2k3k4))
now this is maximized when P is minimal so P=1
it is also maximized when each of k1,k1k2,k1k2k3,k1k2k3k4 are minimized thus
putting this all togeather we get
T=16 and the maximal value is
(1/2 + 1/4 + 1/8 + 1/16)=15/16=0.9375
Posted by Daniel
on 2008-10-03 14:24:29