Introductory Combinatorics
Introductory Combinatorics
5th Edition
ISBN: 9780134689616
Author: Brualdi, Richard A.
Publisher: Pearson,
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Chapter 10, Problem 1E
To determine

To compute: The addition and multiplication table for the integers mod 4.

Expert Solution & Answer
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Answer to Problem 1E

The addition and multiplication table for the integers mod 4 are:

012300123112302230133012 and 012300000101232020230321.

Explanation of Solution

Definition used:

“Let n be a positive integer with n2, then Zn={0,1,,n1}.”

“For any two integers a and b in Zn, ab is the unique remainder when the ordinary sum a+b is divided by n, and ab is the unique remainder when the ordinary product a×b is divided by n.”

Calculation:

The addition and multiplication table for the integers mod 4 is to be calculated.

Thus, n=4.

By the definition of Zn, the set Z4 will have four elements 0, 1, 2 and 3. That is, Z4={0,1,2,3}.

Add 0 with each element of Z4 and obtain the remainder when divided by 4.

00=001=102=203=3

Add 1 with each element of Z4 and obtain the remainder when divided by 4.

10=111=212=313=0

Add 2 with each element of Z4 and obtain the remainder when divided by 4.

20=221=322=023=1

Add 3 with each element of Z4 and obtain the remainder when divided by 4.

30=331=032=133=2

The remainders form the addition table for the integers mod 4 as follows:

012300123112302230133012

Similarly, multiply each elements of Z4 and divided by 4 as follows:

00=001=002=003=0

Multiply 1 with each element of Z4 and obtain the remainder when divided by 4.

10=011=112=213=3

Multiply 2 with each element of Z4 and obtain the remainder when divided by 4.

20=021=222=023=2

Multiply 3 with each element of Z4 and obtain the remainder when divided by 4.

30=031=332=233=1

The remainders form the multiplication table for the integers mod 4 as follows:

012300000101232020230321

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Chapter 10 Solutions

Introductory Combinatorics

Ch. 10 - Prob. 11ECh. 10 - Prob. 12ECh. 10 - Let n = 2m + 1 be an odd integer with m ≥ 2. Prove...Ch. 10 - Use the algorithm in Section 10.1 to find the GCD...Ch. 10 - For each of the pairs of integers in Exercise 14,...Ch. 10 - Apply the algorithm for the GCD in Section 10.1 to...Ch. 10 - Start with the field Z2 and show that x3 + x + 1...Ch. 10 - Does there exist a BIBD with parameters b = 10, v...Ch. 10 - Prob. 19ECh. 10 - Prob. 20ECh. 10 - Determine the complementary design of the BIBD...Ch. 10 - Prob. 22ECh. 10 - How are the incidence matrices of a BIBD and its...Ch. 10 - Show that a BIBD, with v varieties whose block...Ch. 10 - Prove that a BIBD with parameters b, v, k, r, λ...Ch. 10 - Let B be a difference set in Zn. Show that, for...Ch. 10 - Prob. 27ECh. 10 - Show that B = {0, 1, 3, 9} is a difference set in...Ch. 10 - Prob. 29ECh. 10 - Prob. 30ECh. 10 - Prob. 31ECh. 10 - Prob. 32ECh. 10 - Let t be a positive integer. Use Theorem 10.3.2 to...Ch. 10 - Let t be a positive integer. Prove that, if there...Ch. 10 - Assume a Steiner triple system exists with...Ch. 10 - Prob. 36ECh. 10 - Prove that, if we interchange the rows of a Latin...Ch. 10 - Use the method in Theorem 10.4.2 with n = 6 and r...Ch. 10 - Let n be a positive integer and let r be a nonzero...Ch. 10 - Let n be a positive integer and let r and rʹ be...Ch. 10 - Use the method in Theorem 10.4.2 with n = 8 and r...Ch. 10 - Construct four MOLS of order 5. Ch. 10 - Prob. 43ECh. 10 - Construct two MOLS of order 9. Ch. 10 - Prob. 45ECh. 10 - Construct two MOLS of order 8. Ch. 10 - Prob. 47ECh. 10 - Prob. 48ECh. 10 - Prob. 49ECh. 10 - Let A1 and A2 be MOLS of order m and let B1 and B2...Ch. 10 - Construct a completion of the 3-by-6 Latin...Ch. 10 - Prob. 53ECh. 10 - Prob. 54ECh. 10 - Prob. 55ECh. 10 - Prob. 56ECh. 10 - Prob. 57ECh. 10 - Prob. 58ECh. 10 - Prob. 59ECh. 10 - Prob. 60ECh. 10 - Let , where m is a positive integer. Prove that...
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