1. Prove that if $a$ and $b$ are relatively prime to each other, then $\operatorname{gcd}(a,bc) = \operatorname{gcd}(a,c)$.
  2. One of the classic puzzles of antique mathematics was to construct a cube of twice the size of some original cube. Prove that there are no integers $a$ and $b$ such that $2 a^3 = b^3$.
  3. Use the prime factorization theorem to prove that if $\operatorname{gcd}(b,x) = 1$, then $\log_b x$ is irrational.