Exercises for self study. Complete these exercises in the textbook and
compare your solutions against those in the back of the book. These exercises
will not be graded.
p. 259, Exercises 21 and 47
p. 289, Exercise 31
p. 323, Exercise 25
Exercises to be graded. Complete these exercises and turn in your
solutions.
Prove that, for integers $a$, $b$ and $c$, where $a \ne 0$,
(a) if $a \mid b$ and $a \mid c$, then $a \mid (b + c)$,
(b) if $a \mid b$, then $a \mid bc$, and
(c) if $a \mid b$ and $b \mid c$, then $a \mid c$.
Let $m$ be an integer $> 0$. Prove that, for integers $a$ and $b$, if
$a \equiv b \pmod m$, then $a$ mod $m = b$ mod $m$.
Let $m$ be an integer $> 0$. Prove that, for integers $a$ and $b$, if
$a \equiv b \pmod m$, then $\gcd(a, m) = \gcd(b, m)$.
Use the Euclidean algoithm to compute $\gcd(2025, 1823)$ by hand. Show
your work.
Plagiarism and academic dishonesty.
Remember, under any circumstance, you must not copy part or all of another's
work and present it as your own. For more details, read
our course policy on plagiarism and
academic dishonesty.
