Consider a positive integer constant n.
Ten distinct integers are placed on the vertices of a regular decagon satisfying both the following conditions:
• The product of two non adjacent integers on the decagon is a multiple of n.
• The product of any pair of adjacent integers is NOT a multiple of n.
Determine the minimum value of n.
For the decagon an improved minimum is (2*3*5*7*11)^2
The idea is to place five "holes" around the perimeter two spaces apart. Each with no factors of a certain prime and its neighbors with only one factor.
let (a,b,c,d,e) be the number of factors of 2,3,5,7,11 then we have the cycle
(0,2,2,2,2)
(1,1,2,2,2)
(2,0,2,2,2)
(2,1,1,2,2)
(2,2,0,2,2)
(2,2,1,1,2)
(2,2,2,0,2)
(2,2,2,1,1)
(2,2,2,2,0)
(1,2,2,2,1)
You can see that each adjacent pair lacks a sum of 2 in one of the primes, but pairs further apart do sum of at least 2 in all spots.
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Posted by Jer
on 2023-04-25 16:39:11 |
Improvement
