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Complete Residue Systems.pptx
- 1. Complete Residue Systems Jason T. Meregildo MAEd- Mathematics
- 2. What is aCongruence…? •𝒂 ≡ 𝒃 𝒎𝒐𝒅 𝒏 if 𝐧︱𝐚 − 𝐛 •𝒂 − 𝒃 = 𝒌𝒏 ; 𝒌 𝝐 ℝ
- 3. What is a Systemof Residues…? •A subset of ℝ of the set of integers is called a residue system modulo 𝒏 if no two elements of ℝ are congruent 𝐦𝐨𝐝𝐮𝐥𝐨 𝒏.
- 4. •A collection ofn integers 𝒂𝟏, 𝒂𝟐, … , 𝒂𝒏 is a CSR modulo n if every integer is congruent to n to one and only one of the 𝒂𝒌. What is a Complete System of Residues…?
- 5. What is a CompleteSystem of Residues…? 𝒂𝟏, … , 𝒂𝒏 ≡ 𝟎, 𝟏, … , 𝒏 − 𝟏 𝒎𝒐𝒅𝒖𝒍𝒐 𝒏
- 6. Example: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, … , 𝟔
- 7. Solution: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, … , 𝟔 𝑺 = −𝟏𝟐, −𝟒, 𝟏𝟏, 𝟏𝟑, 𝟐𝟐, 𝟖𝟐, 𝟗𝟏
- 8. Solution: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, … , 𝟔 𝑺 = −𝟏𝟐, −𝟒, 𝟏𝟏, 𝟏𝟑, 𝟐𝟐, 𝟖𝟐, 𝟗𝟏
- 9. Solution: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, … , 𝟔 𝑺 = −𝟏𝟐, −𝟒, 𝟏𝟏, 𝟏𝟑, 𝟐𝟐, 𝟖𝟐, 𝟗𝟏
- 10. Solution: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, … , 𝟔 𝑺 = −𝟏𝟐, −𝟒, 𝟏𝟏, 𝟏𝟑, 𝟐𝟐, 𝟖𝟐, 𝟗𝟏
- 11. Solution: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, … , 𝟔 𝑺 = −𝟏𝟐, −𝟒, 𝟏𝟏, 𝟏𝟑, 𝟐𝟐, 𝟖𝟐, 𝟗𝟏
- 12. Solution: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, … , 𝟔 𝑺 = −𝟏𝟐, −𝟒, 𝟏𝟏, 𝟏𝟑, 𝟐𝟐, 𝟖𝟐, 𝟗𝟏
- 13. Solution: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, … , 𝟔 𝑺 = −𝟏𝟐, −𝟒, 𝟏𝟏, 𝟏𝟑, 𝟐𝟐, 𝟖𝟐, 𝟗𝟏
- 14. Example: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔
- 15. Example: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔
- 16. Example: Let 𝒏 =𝟕 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔
- 17. Example 2: Let 𝒏= 𝟒 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑
- 18. Solution: Let 𝒏 =𝟒 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑 𝑺 = 𝟏𝟐, 𝟏𝟏, 𝟖, 𝟑
- 19. Solution: Let 𝒏 =𝟒 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑 𝑺 = 𝟏𝟐, 𝟏𝟏, 𝟖, 𝟑
- 20. Solution: Let 𝒏 =𝟒 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑 𝑺 = 𝟏𝟐, 𝟏𝟏, 𝟖, 𝟑
- 21. Solution: Let 𝒏 =𝟒 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑 𝑺 = 𝟏𝟐, 𝟏𝟏, 𝟖, 𝟑
- 22. Example: Let 𝒏 =𝟒 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑
- 23. Example 2: Let 𝒏= 𝟒 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑
- 24. Solution: Let 𝒏 =𝟒 𝒓 = 𝟎, 𝟏, 𝟐, 𝟑 𝑺 = 𝟏𝟐, 𝟏𝟏, 𝟖, 𝟑
- 25.