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.
n =
6469693230
n is the product of the first ten primes. Each vertex's integer is this product divided by a unique pair of these prime numbers, one member of which is shared with each of its neighboring vertices, as in the following list:
Label divisor
vertex # (quotient) of n
1 111546435 29*2
2 1078282205 2*3
3 431312882 3*5
4 184848378 5*7
5 84021990 7*11
6 45242610 11*13
7 29274630 13*17
8 20030010 17*19
9 14804790 19*23
10 9699690 23*29
For example, neither of the first two vertices is divisible by 2, so their product is also odd; but n is even, so it can't divide into that product.
111546435*1078282205 = 120278535891689175
120278535891689175/6469693230 = 8591323992263512/462120945
=~ 18591072.5
(better approx. is 18591072.49999999891803216147)
n=prod(primes(nthprime(10)))
for i=1:10
prev=mod(i+8,10)+1;
vertex=n/(nthprime(prev)*nthprime(i));
fprintf(' %2d %10d %2d*%-2d\n',i,vertex,nthprime(prev),nthprime(i))
end
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Posted by Charlie
on 2023-04-25 11:50:28 |
I think this is it
