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.
Consider n = p1*p2*p3...p10, where each pi is the i-th prime number.
If we label the vertices of the decagon A, B, C, D, ... J, we can then assign a subset of primes to each vertex such that any two adjacent vertices will be "missing" a factor, while any two nonadjacent vertices contain at least one factor of pi among them. This can easily be done by assigning A = p1*p2*p3*p4...p8, B = p2*p3*p4...p9, and so on, wrapping around such that J = p10*p1*p2*...p7.
Note, for example, that A*J does not contain a factor of p6, and so is not a multiple of n. Meanwhile, B * J contains all factors at least once. Furthermore, since each of A, B, C...J have unique prime factorizations, all of the integers are unique. This implies that the minimum value of n must be less than or equal to the product of the first 10 primes = 6469693230.
I'm not sure if it's possible to do better by using perfect powers of primes instead though...
Edited on April 25, 2023, 10:58 am
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Posted by H M
on 2023-04-25 10:56:25 |
Upper Bound
