a) You are given an ordered list of 7 integers. Prove that there exists a sequence of consecutive terms which sum to a multiple of 7.

b) Given a seven digit number, it is always possible to delete digits from the beginning and/or end (but not the middle) to leave a multiple of 7. For example, given 7654321 we can delete 7, 6 and 1 to obtain 5432 = 7 x 776.

Why is this always possible? Can you find a method to obtain the multiple of 7?

c) Does the result in (b) hold for 6 instead of 7? How about 2017? Describe the set of integers for which the result is true.