## Factor some ideals in ZZ

Find the prime factorizations of the ideals $A = (70)$ and $B = (150)$ in $\mathbb{Z}$. Compute $((70),(150))$.

Evidently $70 = 2 \cdot 5 \cdot 7$ and $150 = 2 \cdot 3 \cdot 5^2$, so that $(70) = (2)(5)(7)$ and $(150) = (2)(3)(5)^2$. Since $\mathbb{Z}$ is a unique factorization domain, $(2)$, $(3)$, $(5)$, and $(7)$ are prime since 2, 3, 5, and 7 are prime.

Now $((70),(150)) = (70,150) = (10)$, since $70 = 7 \cdot 10$, $150 = 15 \cdot 10$, and $10 = 150 - 2 \cdot 70$.