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Tim from Gravesend Grammar School sent in this solution. Well done Tim.
Let the original number chosen be n.
The new six digit number formed is made by adding n to n shifted three digits to the left, ie 1000n, so the new number is 1000n + n, or 1001n. 1001=7*11*13, so 1001n will always have the factors 7, 11 and 13