1: Computing using modular arithmetic. Compute the value of the following expressions: (a) 387 mod 3 (b) (72 (65) + 211) mod 7 (c) (77 (65) + 147) mod 7 (d) 4412 mod 6 (e) 1712 mod 3 2: Factoring out the gcd. Suppose that for positive integers, a and b, gcd(a,b) = d. What is gcd(a/d, b/d)?Justify your answer. 3: Largest number with a specified number of digits. (a) What is the decimal representation of (10000)7? (b) What is the largest number that can be represented with four digits in base 7? (Give the base-7 representation of the number as well as its decimal representation.) (c) What is the relationship between the values of the two numbers in the previous two questions? 4: Choosing a lineup for a traveling basketball team. There are 20 members of a basketball team. (a) The coach must select 12 players to travel to an away game. How many ways are there to select the players who will travel? (b) From the 12 players who will travel, the coach must select her starting line-up. She will select a player for each of the five positions: center, right forward, left forward, right guard, left guard. How many ways are there for her to select the starting lineup? (c) From the 12 players who will travel, the coach must select her starting line-up. She will select a player for each of the five positions: center, right forward, left forward, right guard, left guard. However, there are only three of the 12 players who can play center. Otherwise, there are no restrictions. How many ways are there for her to select the starting line-up? 5: Assigning summer camp activities. A camp offers 4 different activities for an elective: archery, hiking, crafts and swimming. The capacity in each activity is limited so that at most 35 kids can do archery, 20 can do hiking, 25 can do crafts and 20 can do swimming. There are 100 kids in the camp. How many ways are there to assign the kids to the activities? 6: Distributing prizes. Ten prizes are given to a class with 100 students. Each student can receive at most one prize. Alice and Bob are two students in the class. (a) If the prizes are identical, how many ways are there to distribute the prizes so that either Alice or Bob (or both) receive a prize? (b) If the prizes are different, how many ways are there to distribute the prizes so that either Alice or Bob (or both) receive a prize