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- Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .An n×n magic square is a square array of numbers (n > 1) such that the sum of every row, of every column, and of the two diagonals is the same. For which n is there a magic square whose entries are consecutive positive integers with a sum of 2015? For each such n, what are the largest and smallest entries in the square? Prove that (1,1) is an element of largest order in Zn1 + Zn2. State the general case. Needs Complete solution with 100 % accuracy.
- Use Fermat's theorem to show that 1n + 2n + 3n + 4n is divisible by 5 if and only if n is not divisibleby 4.Question 1. Show that if seven integers are selected from the first 10 positive integers,there must be at least two pairs of these integers with the sum 11.Question 2. Is the conclusion still true if six integers are selected rather thanseven in number 1 question?How many elements of order 4 does Z4 ⊕ Z4 have? (Do not do this by examining each element.) Explain why Z4 ⊕ Z4 has the same number of elements of order 4 as does Z8000000 ⊕ Z400000. Generalize to the case Zm ⊕ Zn.
- Consider the following recurrence relation. an =8an-1 -15an-2 +4n/4 with boundary conditions as a0=−3 and a1=2. 1. What is the order of the above recurrence relation? 2. What is the characteristic equation from the homogeneous part of the given recurrence relations? 3.What is the guessed solution of the particular part? 5. What is the final solution of the recurrence relationThis question is related to unique factorisation in the Gaussian integers. In the Gaussian integers, find the gcd of 13 and 8 + i. The problem says "the gcd" (which is correct), but what are all the possibilities for the gcd?What is the characteristic polynomial for the recurrence relation an = 50p-1 - 7.?
- an =8an-1 -15an-2 +4n/4 with boundary conditions as a0=−3 and a1=2. What is the final solution of the recurrence relation?A knight on a chessboard can move one space horizontally (in either direction) and two spaces vertically (in either direction) or two spaces horizontally (in either direction) and one space vertically (in either direction). Suppose that we have an infinite chessboard, made up of all squares (m, n), where m and n are nonnegative integers that denote the rownumber and the column number of the square, respectively. Use mathematical induction to show that a knight starting at (0, 0) can visit every square using a finite sequence of moves. [Hint: Use induction on the variable s = m + n.]5.2.8 suppose that a store offers gift certificates and denominations of $25 and $40. Determine the possible total amounts you can form using these gift certificates. Prepare answer using strong induction.
