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Navigating the grid

Using the number grid shown (A) it is possible to generate nine-digit numbers in the following way: Start on any square and then move horizontally, vertically or diagonally to an adjacent square that hasn’t already been visited. Repeat until all nine squares have been visited. For example, the path shown in diagram B generates the number 235968741.

I have listed all the numbers that can be generated in this way, and that end in a certain digit. Some of these are divisible by four (and only four) of the numbers 1 to 9. The rest are divisible by five (and only five) of the numbers 1 to 9.

How many numbers are there in my list?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 19 June. The Editor’s decision is final. Please send entries to Enigma 1750, New Å®ÉúСÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1744 Clive’s number: CLIVE is 37125

The winner Simon Armstrong of Guisborough, Redcar and Cleveland, UK