1.

Find the greatest common divisor of 471 and 564 using the Euclidean Algorithm and then find integers $r$ and $s$ such that $\gcd(471,564) = 471r+564s\text{.}$

2.

Find a single integer $n$ such that the ideal $\langle n \rangle$ is the smallest ideal in $\Z$ containing both $471$ and $564\text{.}$

3.

In the quotient ring $\Z/\langle 564 \rangle\text{,}$ find an element $a + \langle 564\rangle$ such that $(a+\langle 564\rangle)(471 + \langle 564\rangle) = 3 + \langle 564 \rangle\text{.}$

4.

Is $471 + \langle 564\rangle$ a unit in $\Z/\langle 564\rangle\text{?}$ Explain.

5.

In $\Q[x]\text{,}$ find the greatest common divisor of the polynomials $a(x) = x^3 + 1$ and $b(x) = x^4 + x^3 + 2x^2 + x - 1\text{.}$ Then express the gcd as a combination of the two polynomials (as in Bezout's lemma).

6.

Find the greatest common divisor of $x^{24}-1$ and $x^{15}-1$ in $\Q[x]\text{,}$ and then express the gcd as a combination of the two polynomials.

7.

Find a coset $a(x) + \langle x^{24}-1\rangle$ of $\Q[x]/\langle x^{24}-1\rangle$ such that $(a(x) + \langle x^{24}-1\rangle)(x^{15}-1) + \langle x^{24}-1\rangle) = x^3-1 + \langle x^{24}-1\rangle\text{.}$