Problems:
For each topic, demonstrate a knowledge and capability by giving the following information:
1) Problem Solution: (solution for an even number problem) see below...
2) Personal Observation: (personal comment on the topic including advice to others on how to study and understand it).
Algorithms
3. Describe an algorithm that takes as input a list of n integers and produces as output the largest difference between consecutive integers in the list.
Integers
4. What is the greatest common divisors of these pairs of integers?
a) 22 * 33 * 55, 25 * 33 * 52
b) 2 * 3 * 5 * 7 * 11 * 13, 211 * 39 * 11 * 1714
c) 17, 1717
d) 22 * 7, 53 * 13
e) 0, 5
f) 2 * 3 * 5 * 7, 2 * 3 * 5 * 7
Number Theory
5. Find all solutions to the system of congruences.
x = 2(mod 3)
x = 1(mod 4)
x = 3(mod 5)
Matrices
6. The n x n matrix A = [aij] is called a diagonal matrix if aij = 0 when i ≠ j. Show that the product of two n x n diagonal matrices is again a diagonal matrix. Give a simple rule for determining this product.