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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120で割って119余る数

{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}

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

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60で割って58余る数

{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}

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

{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}

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

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40で割って27余る数

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

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

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

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

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

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

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

{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}

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

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

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30で割って29余る数

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

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

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

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

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

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

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

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

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

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

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

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

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20で割って7余る数

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

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

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

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

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

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

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

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

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

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

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

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

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15で割って8余る数

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

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

{21,22,24,25,26,28,30}

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

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

{51,52,54,55,56,57,60}

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

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

{81,82,84,85,86,88,90}

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

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

{111,112,114,115,116,117,120}

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3の倍数

{01,02,04,05,10}

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

{22,25,26,28}

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

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

{52,55,56}

{61,62,64,65,70}

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

{82,85,86,88}

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

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

{112,115,116}

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4の倍数

{01,02,05,10}

{11,13,14,17,19}

{22,25,26}

{31,34,35,37}

{41,43,46,49,50}

{55}

{61,62,65,70}

{71,73,74,77,79}

{82,85,86}

{91,94,95,97}

{101,103,106,109,110}

{115}

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5の倍数

{01,02}

{11,13,14,17,19}

{22,26}

{31,34,37}

{41,43,46,49}

{}

{61,62}

{71,73,74,77,79}

{82,86}

{91,94,97}

{101,103,106,109}

{}

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6で割って1余る数

{02}

{11,14,17}

{22,26}

{34}

{41,46}

{}

{62}

{71,74,77}

{82,86}

{94}

{101,106}

{}

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8で割って2余る数

{}

{11,14,17}

{22}

{}

{41,46}

{}

{62}

{71,77}

{86}

{94}

{101,106}

{}

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10で割って1余る数

{}

{14,17}

{22}

{}

{46}

{}

{62}

{77}

{86}

{94}

{106}

{}

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残りは12のみだがうまくいかない

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