Compute o1 (100), 02 (750), and 03(1024).

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Recall the Modular Equivalence Theorem from Chapter 8: a = b (mod d) means that a and b have the same remainder when divided by d. Furthermore, note that if an integer n has a remainder of zero when divided by d, then d divides n. Let n be a positive integer. The decimal representation of n is an expression of the form n = d,10k + dx-110"-1 + ... + d, 10' + d,10° Where the coefficients dr, ., do are called the digits of n. The digits take on the values 0, 1,2, .,9 and the first digit d, is nonzero. .... a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the textbook. b) Use part a to prove that a positive integer is divisible by 9 if and only if the sum of its digits dy + ...+ d, is divisible by 9.

Use the factorization A = PDP-1 to compute Ak, where k represents an arbitrary positive integer. 2-142 1³:1 = 01 AK = - 15 72 -4 19 49 30 -2 9 1 -4

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The following bit pattern represents a floating point number in IEEE 754 single precision format: 1 10000011 10100000000000000000000 The value of the number in decimal form is

1) Write as single cycle or product of disjoint cycles: (1234) (245)(1534)(513)

Each character in a password is either a digit [0-9] or lowercase letter [a-z]. How many valid passwords are there with the given restriction(s)? The length is 17.