Exercise · ZW-COMPLETE-RESIDUE-SYSTEM-008 · ★★☆ Intermediate
Let \(m=8\) and consider \(\{1,3,5,7\}\) as a set of residues. How many integers are needed to complete a complete residue system?
Choose one option.
Show answer
- B. \(4\)
Explanation
There are \(8\) classes and the set covers \(4\) (the odd ones), so \(4\) even classes are missing. Answering \(0\) confuses the number of elements with the number of covered classes.
Review the concept: Complete residue system
A complete residue system modulo \(m\) is a set of \(m\) integers that contains exactly one representative from each congruence class modulo \(m\).
\[\{r_1,\dots,r_m\}\ \text{with}\ r_i\not\equiv r_j\ (\bmod\ m)\ \forall i\neq j\]
More practice problems: complete residue system
- ZW-COMPLETE-RESIDUE-SYSTEM-001 How many elements must a complete residue system modulo 7 have?
- ZW-COMPLETE-RESIDUE-SYSTEM-002 Which of these sets is a complete residue system modulo 5?
- ZW-COMPLETE-RESIDUE-SYSTEM-003 How many congruence classes are there modulo 9?
- ZW-COMPLETE-RESIDUE-SYSTEM-004 If \0,1,2,3\ is complete modulo 4, what is the remainder of 7 divided by 4?
- ZW-COMPLETE-RESIDUE-SYSTEM-005 Which of these integers belongs to the class of 2 modulo 6?
- ZW-COMPLETE-RESIDUE-SYSTEM-006 Determine whether \0,1,7,3,4,5,6\ is a complete residue system modulo 7 and explain why.
- ZW-COMPLETE-RESIDUE-SYSTEM-007 If \1,2,3,4,5,6\ is complete modulo 6, which of these sets is also complete modulo 6?
- ZW-COMPLETE-RESIDUE-SYSTEM-009 Prove that if gcd(a,7)=1 and \r_1,dots,r_7\ is complete modulo 7, then \ar_1,dots,ar_7\ is also com…
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