Exercise · ZW-COMPLETE-RESIDUE-SYSTEM-006 · ★★☆ Intermediate
Determine whether \(\{0,1,7,3,4,5,6\}\) is a complete residue system modulo \(7\) and explain why.
Choose one option.
Show answer
- D. No, because \(7\equiv0\) repeats the class of \(0\)
Explanation
Since \(7\equiv0\pmod 7\), the set contains two elements of the class of \(0\) and misses the class of \(2\). Counting seven elements is not enough.
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-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-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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