Maths Item of the Month – March 2013
Next Factor
Starting with the number 2, integers are added, in order, as vertices to a graph so that any
two vertices are joined if one of them is a factor of the other. A graph for {2,3, 4,5,6} is
shown.
What is the maximum integer that can be reached if none of the edges of the graph are
allowed to cross?
This is a version of the nrich problem Factors and Multiples Graphs.
For more problems see the nrich website.
Solution
Firstly, we can show the graph can be drawn up to and including the number 23 .
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The same does not hold true of the graph containing 24. If we examine the graph for the
integers {2,3,6,12, 24} , we see that they almost form the complete graph, K 5 .
The complete graph K 5 is important, as any complete graph with 5 vertices or more must
contain a crossing. The above graph is not complete, as the vertices at 2 and 3 are not
joined. However, if we draw the graph for {2,3,6,7,12,14, 21, 24} , we can ‘create’ an indirect
path between them.
As such we can see that the maximum integer that can be reached is 23, as any graph after
this will contain a complete graph with 5 vertices.
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