1. If n=4k+3, does 8 divide n -1 ? Give a proof for your answer. 2. Represent the decimal number 1217 in binary notation. Show work. 3. Represent the binary number 11011111 as a decimal number. Show work. \f\f5. If a and b are positive integers, show that god(a, b) = god(a, a + b). 6. In how many ways can we select a committee of four Republicans, three Democrats, and two Independents from a group of 10 distinct Republicans, 12 distinct Democrats, and four distinct Independents? 7. In how many ways can 10 distinct books be divided among three students if the first student gets five books, the second three books, and the third two books?8. In the Maryland Lotto game, to win the grand prize the contestant must match six distinct numbers 1 through 49 randomly drawn by a lottery representative. What is the probability of choosing the winning numbers?9. Suppose there is a family with four children. Assume that it is equally probable for a boy or a girl to be born. a. What is the probability of all girls? b. What is the probability of all girls given there is at least one girl? c. What is the probability of at least one boy and one girl? d. Are the events \"there are children of both sexes\" and \"there is at most one boy\" independent