Exercise · ZW-COMPLETE-RESIDUE-SYSTEM-007 · ★★☆ Intermediate
If \(\{1,2,3,4,5,6\}\) is complete modulo \(6\), which of these sets is also complete modulo \(6\)?
Choose one option.
Show answer
- C. \(\{2,3,4,5,6,7\}\)
Explanation
The set \(\{2,3,4,5,6,7\}\) has remainders \(2,3,4,5,0,1\), all distinct. The others repeat classes: \(11\equiv5\), \(6\equiv0\), or the even numbers repeat.
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-008 Let m=8 and consider \1,3,5,7\ as a set of residues. How many integers are needed to complete a com…
- 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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