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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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a. Suppose a 1 , a 2 , ... , a n is a sequence of n integers none of which is divisible by n. Show that at least one of the differences a i a j (for i j ) must be divisible by n.
b. Show that every finite sequence x 1 , x 2 , ... , x n of n integers has a consecutive subsequence x i + 1 , x i + 2 , ... , x j 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 determine

(a)

To show that at least one difference aiaj(ij) must be divisible by n.

Explanation

Given information:

Suppose a1,a2,....,an is a sequence of n integers none of which is divisible by n.

Calculation:

Totally there are n integers and these are to be divided by n. So this is very much clear that there will be n1 possible remainders. Now according to the pigeonhole principle there will be at least two integers say ai and aj where ij from the sequence a1,a2,....,an will have the same remainder.

Now suppose there are two integers ai and aj which have the remainder rk,1rkn1

To determine

(b)

To show that every finite sequence x1,x2......xn of n integers has a consecutive subsequence xi+1,xi+2,.........,xj whose sum is divisible by n.

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