Exercise · ZW-EUCLIDEAN-DOMAIN-001 · ★☆☆ Basic
In \(\mathbb{Z}\) with \(N(n)=|n|\), which quotient \(q\) and remainder \(r\) satisfy \(17=5q+r\) with \(0\le r<5\)?
Choose one option.
Show answer
- C. \(q=3,\ r=2\)
Explanation
Since \(5\cdot3=15\), we get \(r=2\), and \(0\le2<5\). The pair with \(q=2\) has remainder 7, which is not less than 5; the other two do not add up to 17.
Review the concept: Euclidean domain
An integral domain \(D\) with a norm \(N\) (a function from \(D\setminus\{0\}\) to the nonnegative integers) allowing division of \(a\) by \(b\neq0\) with quotient \(q\) and remainder \(r\) zero or of smaller norm than \(b\).
\[a=bq+r,\qquad r=0\ \text{ or }\ N(r)
More practice problems: euclidean domain
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- ZW-EUCLIDEAN-DOMAIN-008 If D is a Euclidean domain, what can be said about its ideals?
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