Algebra - Durbin, John, R. Modern Algebra

12.12. Prove that if $a$ dan $b$ are integers, not both zero, then there are infinitely many pairs of integers $m,n$ such that $(a,b)=am+bn$.

12.13. Prove that if $c$ is a positive integer, then $(ac,bc)=(a,b)\cdot c$.

12.15. Prove that if $p$ is a prime and $a$ is an integer, and $a$ is not divisible by p, then $(a,p)=1$.

12.22. Prove that if $a,b,c$ are integers, not all zero, then they have a greatest common divisors, which can be written as a linear combination of $a,b,$ and $c$.

13.16. Prove that if $n$ is an integer, then $\sqrt{n}$ is rational iff $n$ is a perfect square.

13.19. Prove that if $a$ and $b$ are positive integers, then
$$(a,b)[a,b]=ab$$
when [a,b] is least common multiple.

Some Notes on Root of Unity

Let $a_,\dots,a_n$ be positive real numbers and let $\omega$ be a primitive nth root of unity. If the sides of an equiangular polygon have lengths $a_,\dots,a_n$ (in counterclockwise order) then
$$1+\omega+\omega^2+\dots+\omega^{n-1}=0$$;
$$a_1 + a_2 \omega +a_3 \omega ^2+ \dots + a_n \omega^{n-1}=0$$.