Differences
Pick three test values - calculate differences (why are only three listed?) - and then the product. Can you find an exception? The question is "Why not?".
The second half of the question is about multiples of three. Can you 'categorise' numbers in terms of their relationship to mulltiples of three and how does this help?
Have you seen the problem Take Three from Five ?
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Adding All Nine
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!
Repeaters
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.