Download Name: Math 325 Homework 1 Question: 1 2 3 Total Points: 21 6 8 35

Name:
Math 325 Homework 1
Question:
1
2
3
Total
Points:
21
6
8
35
Score:
1. (21 points) Label each statement below as true or false. If the statement is true, prove it. If
the statement is false, give an example that shows it is false (i.e. a counterexample). (Note on
how to correctly write counterexamples: if you want to show that a statement of the form “If P
then Q” is false, you need to give an example of something that makes P true and Q false. For
example, consider the statement “If a|b then b|a.” This statement is false. Here is one correct
way to show that it is false: consider a “ 2 and b “ 4. Then a|b since 2|4, but bffla since 4ffl2.)
All parts of this problem must be written in complete sentences. Please do not use symbols
(such as ñ, 6, etc.); use words. (You may, however, use the symbol for divides, but sentences
should always start with a word, not a symbol or number. So, instead of writing “2|4.” try
writing something like “Note that 2|4.”)
(a) If a|c and b|c, then ab|c.
(b) If a|b and a|c, then a2 |bc.
(c) If a|bc then a|b or a|c.
2. (6 points) Use the Euclidean Algorithm to find the greatest common divisor of the following
pairs of numbers.
(a) 12345 and 67890
(b) 272, 1479
3. (8 points)
(a) Compute gcdpa, a ` 1q for three different values of a.
(b) Compute gcdpa, a ` 2q for four different values of a. (Try to get as many different values
for gcdpa, a ` 2q as you can.)
(c) compute gcdpa, a ` 3q for four different values of a. (Try to get as many different values
for gcdpa, a ` 3q as you can.)
(d) Compute gcdpa, a ` 6q for four different values of a. (Try to get as many different values
for gcdpa, a ` 6q as you can.)
(e) Based on your answers to the above, what do you think is true about gcdpa, a ` nq in
general? (Test your conjecture for some new values of a and n and revise your conjecture
if necessary.)