Exercise · ZW-UNIQUE-FACTORIZATION-DOMAIN-010 · ★★★ Advanced
In \(\mathbb{Z}\), how many positive divisors does 360 have, using its factorization \(2^3\cdot3^2\cdot5\)?
Choose one option.
Show answer
- B. 24
Explanation
Each positive divisor chooses an exponent from 0 to 3 for 2, from 0 to 2 for 3 and from 0 to 1 for 5, thanks to the uniqueness of factorization: \(4\cdot3\cdot2=24\).
Review the concept: Unique factorization domain
An integral domain \(D\) in which every nonzero nonunit element \(a\) is a product of irreducibles \(p_i\), and this factorization is unique up to order and associates.
\[a=p_1p_2\cdots p_r\]
More practice problems: unique factorization domain
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