Extra Problems

**A**
(correct) if the
statement and proof are correct. Give this grade even if the
given proof
is not the simplest possible, or is not the one you would have given.

**C** (partially correct) if the
statement is correct and the proof is largely valid, but contains a few
statements
that are incorrect or insufficiently justified.

**F** (incorrect) if the statement is
incorrect, or the main idea of the proof is incorrect or has faulty
logic, or
if most of the statements in the proof are incorrect.

Be sure to justify grades of C or F.

Most of these problems, and the concept for this type of problem, are taken from

Section 1.1

x1.1.1 If it is

x1.1.2 (ptg) Statement: There is a unique set of three consecutive odd integers that are all prime.

Proposed Proof: The consecutive odd integers 3, 5, and 7 are all prime. Suppose that

Section 1.2

x1.2.1 (ptg)
Statement: Suppose *a*, *b*,
and *c* are integers. If *a* | *b* and *a*
| *c*,
then *a* | (*b*+*c*).

Proposed Proof: Since *a* | *b* we know that *b*
= *aq* for some integer *q*.
Similarly, since
*a* | *c* we know that *c* = *aq*
for some integer *q*. Then *b + c* = *aq**
+ aq *= 2*aq *= *a*(2*q*).
This shows that *a | *(*b + c*).

Proposed Proof: 1 |