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Same remainder when divided by 3

Source:

September 5, 2010

modular arithmeticnumber theory unsolvednumber theory

Problem Statement

Let

x_1, x_2,\cdots, x_n

be

n

integers. Let

n = p + q

, where

p

and

q

are positive integers. For

i = 1, 2, \cdots, n

, put

S_i = x_i + x_{i+1} +\cdots + x_{i+p-1} \text{ and } T_i = x_{i+p} + x_{i+p+1} +\cdots + x_{i+n-1}

(it is assumed that

x_{i+n }= x_i

for all

i

). Next, let

m(a, b)

be the number of indices

i

for which

S_i

leaves the remainder

a

and

T_i

leaves the remainder

b

on division by

3

, where

a, b \in \{0, 1, 2\}

. Show that

m(1, 2)

and

m(2, 1)

leave the same remainder when divided by

3.

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