is a sequence of n integers none of which is divisible by n. Show that at least one of the differences
) must be divisible by n.
b. Show that every finite sequence of n integers has a consecutive subsequence whose sum is divisible by n. (For instance, the sequence 3, 4, 17, 7, 16 has the consecutive subsequence 17, 7, 16 whose sum is divisible by 5.) (From: James E. Schultz and William F. Burger, “An Approach to Problem-Solving Using Equivalence Classes Modulo n,” College Mathematics Journal (15), No. 5, 1984, 401-405.
To show that at least one difference must be divisible by .
Suppose is a sequence of integers none of which is divisible by .
Totally there are integers and these are to be divided by . So this is very much clear that there will be possible remainders. Now according to the pigeonhole principle there will be at least two integers say and where from the sequence will have the same remainder.
Now suppose there are two integers and which have the remainder
To show that every finite sequence of integers has a consecutive subsequence whose sum is divisible by .
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