1. (4pts each) What is

a. gcd(140,100)

b. lcm(140,100)

c. 13 mod 3

d. 26 mod13

e. 0 mod 8

2. (30pts) Prove that
Please explain each step

fully.

3. (6pts) Assume there are 10 graduate students and 20
professors. The two groups

are disjoint.

How ways to

a. Staff a class with one professor and one grad student?

b. Staff a class with one instructor who is either a professor or grad student?

c. Staff the class with one professor and 3 grad students?

4. (9pts)What is the Pigeon Hole Principle? (Nothing complex please)

5. Persons A,B,C,D,E,F are to be seated in a row of six
chairs. How many ways are

there to

a. (5pts) seat them all?

b. (10pts )seat them if A and B must sit next to each other?

6. Given an 8 bit string.

a. How many way strings are there.

b. How many strings start with 11 or 00

c. How many strings start with 00 or ends with a 11.