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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 .a. Excluding the identity cipher, how many different translation ciphers are there using an alphabet of n "letters"? b. Excluding the identity cipher, how many different affine ciphers are there using an alphabet of n "letters," where n is a prime?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?
- Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in n!, same with (n-r)! and r!. Then cancel numbers appropriately in the numerator and denominator and simplify to one number.1. Suppose we build an LFSR machine that works mod 3 instead of mod 2. It uses a recurrence of length 2 of the form Xn+2 = CoXn+ c1xn+1__mod 3 to generate the sequence 1, 1, 0, 2, 2, 0, 1, 1. Set up and solve the matrix equation to find the coefficients co and c1.expand 7q(q-2)
- Use the characteristic root technique to solve the following recurrence relations ( find a closed-form expression): a) an= 7an-1 - 10an-2 with a0 = 2 a1 = 3 b) an= 3an-1 + 4an-2 with a0 = 5 a1 = 820. Let d = (826, 1890). Use the Euclidean algorithm to compute d, then express d as a linear combination of 826 and 1890.74. P is a natural number of at least 6 digits and its left- most digit is 7. When this leftmost digit is removed from P, the number thus obtained is found to be 1/21 times of P. What is the product of the all the nonzero digits of P.
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