120＝2^3・3・5の約数は自然数を覆う剰余系を構成する。

x=ai mod(ni)

(ai,ni)

n1＜n2＜・・・、ここではn1=3とする

エルデシュは1より大きい210の真の約数であるような法niに対しては常に被覆系ができることをしめした

ai: 0,0,0,1,0, 1, 1, 2, 2,23, 4, 5, 59,104

ni: 2,3,5,6,7,10,14,15,21,30,35,42,70,105

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3以上の約数は

3，4,5,6,8,10,12,15,20,24,30,40,60,120

{01,02,03,04,05,06,07,08,09,10}

{11,12,13,14,15,16,17,18,19,20}

{21,22,23,24,25,26,27,28,29,30}

{31,32,33,34,35,36,37,38,39,40}

{41,42,43,44,45,46,47,48,49,50}

{51,52,53,54,55,56,57,58,59,60}

{61,62,63,64,65,66,67,68,69,70}

{71,72,73,74,75,76,77,78,79,80}

{81,82,83,84,85,86,87,88,89,90}

{91,92,93,94,95,96,97,98,99,100}

{101,102,103,104,105,106,107,108,109,110}

{111,112,113,114,115,116,117,118,119,120}

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3の倍数

{01,02,04,05,07,08,10}

{11,13,14,16,17,19,20}

{22,23,25,26,28,29}

{31,32,34,35,37,38,40}

{41,43,44,46,47,49,50}

{52,53,55,56,58,59}

{61,62,64,65,67,68,70}

{71,73,74,76,77,79,80}

{82,83,85,86,88,89}

{91,92,94,95,97,98,100}

{101,103,104,106,107,109,110}

{112,113,115,116,118,119}

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5の倍数

{01,02,04,07,08}

{11,13,14,16,17,19}

{22,23,26,28,29}

{31,32,34,37,38}

{41,43,44,46,47,49}

{52,53,56,58,59}

{61,62,64,67,68}

{71,73,74,76,77,79}

{82,83,86,88,89}

{91,92,94,97,98}

{101,103,104,106,107,109}

{112,113,116,118,119}

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6で割って1余る数

{01,02,04,08}

{11,14,16,17}

{22,23,26,28,29}

{32,34,38}

{41,44,46,47}

{52,53,56,58,59}

{62,64,68}

{71,74,76,77}

{82,83,86,88,89}

{92,94,98}

{101,104,106,107}

{112,113,116,118,119}

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10で割って1余る数

{02,04,08}

{14,16,17}

{22,23,26,28,29}

{32,34,38}

{44,46,47}

{52,53,56,58,59}

{62,64,68}

{74,76,77}

{82,83,86,88,89}

{92,94,98}

{104,106,107}

{112,113,116,118,119}

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15で割って2余る数

{04,08}

{14,16}

{22,23,26,28,29}

{34,38}

{44,46}

{52,53,56,58,59}

{64,68}

{74,76}

{82,83,86,88,89}

{94,98}

{104,106}

{112,113,116,118,119}

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30で割って4余る数

{08}

{14,16}

{22,23,26,28,29}

{38}

{44,46}

{52,53,56,58,59}

{68}

{74,76}

{82,83,86,88,89}

{98}

{104,106}

{112,113,116,118,119}

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残りは4,8,12,20,24,40,60,120

４の倍数

{}

{14}

{22,23,26,29}

{38}

{46}

{53,58,59}

{}

{74}

{82,83,86,89}

{98}

{106}

{113,118,119}

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残りは8,12,20,24,40,60,120

8で割って2余る数

{}

{14}

{22,23,29}

{38}

{46}

{53,58,59}

{}

{}

{83,86,89}

{}

{}

{113,118,119}

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残りは12,20,24,40,60,120

12で割って11余る数

{}

{14}

{22,29}

{38}

{46}

{53,58}

{}

{}

{86,89}

{}

{}

{113,118}

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残りは20,24,40,60,120

20で割って18余る数

{}

{14}

{22,29}

{}

{46}

{53}

{}

{}

{86,89}

{}

{}

{113}

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残りは24,40,60,120

24で割って22余る数

{}

{14}

{29}

{}

{}

{53}

{}

{}

{86,89}

{}

{}

{113}

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残りは40,60,120・・・うまくいかない

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