MATH 2345 section 54/55
Homework 11
Solutions
Q1. Prove that the product of any two rational numbers is a rational number. (Direct)
$
$
∎
Q2. Prove that for all integers
࠵?
and
࠵?
, if
࠵?|࠵?
then
࠵?
5
|࠵?
5
. (Direct)
Proof. Let
࠵?
and
࠵?
be [particular but arbitrarily chosen] integers and
࠵?|࠵?
. We must show that
࠵?
5
|࠵?
5
. Since
࠵?|࠵?
, by definition of divisibility,
࠵? = ࠵?. ࠵?
for some integer
࠵?
. Raise both sides to
power of 2. We have
࠵?
5
= ࠵?
5
. ࠵?
5
. Since
࠵?
5
is an integer (it is a product of two integers), we have
that
࠵?
5
= ࠵?
7
. ࠵?
5
for some integer
࠵?′
. By definition of divisibility, we have
