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.

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