1. How Many solutions (x,y) are there to the equation 53x + 26y = 88 mod 97? 2. Find x if 3*x = 4 mod 7. 3. Find all x such that 2x = 4 mod 6 4. List all lines which go through (0,3) and (2,3) mod 6. 4. List all points on the line x =3 mod 5. 5. List all the lines in mod 7 which pass through the point (2,3). 6. Suppose a tournament has 24 players and 36 games, with each player playing in 6 games, and each game containing an equal number of players. How many players are in each game. 7. Suppose x^4 + x^3 +1 is irreducible in mod 2. Let r be a root of this polynomial. a. How large is the new field when we include t. b. Compute t^3 * (t^2+t+1) in the field. c. How many lines are there in this new field? How many points on a line? 8. How many lines are there in mod 37? 9. Show x^2 +2 is irreducible mod 5. 10. Suppose we construct a tournament with 25 players with C total contests, and 5 players in each contest. Each player, and each pair of players appears in the same number of contests. Explain why each player is in C/5 contests, and each pair is in C/30 contests.