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

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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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4の倍数

{01,02,05,07,10}

{11,13,14,17,19}

{22,23,25,26,29}

{31,34,35,37,38}

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

{53,55,58,59}

{61,62,65,67,70}

{71,73,74,77,79}

{82,83,85,86,89}

{91,94,95,97,98}

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

{113,115,118,119}

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5の倍数

{01,02,07}

{11,13,14,17,19}

{22,23,26,29}

{31,34,37,38}

{41,43,46,47,49}

{53,58,59}

{61,62,67}

{71,73,74,77,79}

{82,83,86,89}

{91,94,97,98}

{101,103,106,107,109}

{113,118,119}

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6の倍数はない

6で割って1余る数

{02}

{11,14,17}

{22,23,26,29}

{31,34,38}

{41,46,47}

{53,58,59}

{62}

{71,74,77}

{82,83,86,89}

{94,98}

{101,106,107}

{113,118,119}

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8の倍数はない

8で割って1余る数

{02}

{11,14}

{22,23,26,29}

{31,34,38}

{46,47}

{53,58,59}

{62}

{71,74,77}

{82,83,86}

{94,98}

{101,106,107}

{118,119}

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10の倍数はない

10で割って1余る数

{02}

{14}

{22,23,26,29}

{34,38}

{46,47}

{53,58,59}

{62}

{74,77}

{82,83,86}

{94,98}

{106,107}

{118,119}

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12の倍数はない

12で割って1余る数はない

12で割って2余る数

{}

{}

{22,23,29}

{34}

{46,47}

{53,58,59}

{}

{77}

{82,83}

{94}

{106,107}

{118,119}

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15の倍数はない

15で割って1余る数

{}

{}

{22,23,29}

{34}

{47}

{53,58,59}

{}

{77}

{82,83}

{94}

{107}

{118,119}

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20の倍数はない

20で割って1余る数はない

20で割って2余る数

{}

{}

{23,29}

{34}

{47}

{53,58,59}

{}

{77}

{83}

{94}

{107}

{118,119}

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24の倍数はない

24で割って1余る数はない

24で割って2余る数はない

24で割って3余る数はない

24で割って4余る数はない

24で割って5余る数

{}

{}

{23}

{34}

{47}

{58,59}

{}

{}

{83}

{94}

{107}

{118,119}

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30の倍数はない

30で割って1余る数はない

30で割って2余る数はない

30で割って3余る数はない

30で割って4余る数

{}

{}

{23}

{}

{47}

{58,59}

{}

{}

{83}

{}

{107}

{118,119}

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40の倍数はない

40で割って1余る数はない

40で割って2余る数はない

40で割って3余る数

{}

{}

{23}

{}

{47}

{58,59}

{}

{}

{}

{}

{107}

{118,119}

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うまくいかないが、一つの約数に対して(ai,ni)のaiは一つと限らないと剰余系の意味がない。

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