{"id":5262,"date":"2021-05-17T15:48:10","date_gmt":"2021-05-17T06:48:10","guid":{"rendered":"https:\/\/math-travel.com\/?p=5262"},"modified":"2026-02-11T16:06:53","modified_gmt":"2026-02-11T07:06:53","slug":"euclidean-algorithm","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-a\/euclidean-algorithm\/","title":{"rendered":"\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u3084\u308a\u65b9\u3068\u8a3c\u660e\uff1a\u6700\u5927\u516c\u7d04\u6570\u3092\u4e00\u77ac\u3067\u6c42\u3081\u308b\u624b\u9806\u3092\u89e3\u8aac"},"content":{"rendered":"\n
\u300c\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u3084\u308a\u65b9\u304c\u5206\u304b\u3089\u306a\u3044\u300d<\/span> \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u304c\u5168\u7136\u3067\u304d\u306a\u304f\u3066\u2026<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306f\u30012\u3064\u306e\u6574\u6570\u306e\u6700\u5927\u516c\u7d04\u6570<\/span>\u3092\u6c42\u3081\u308b\u3068\u304d\u306b\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n\n \u6700\u5927\u516c\u7d04\u6570\u306e\u6c42\u3081\u65b9\u306f\u975e\u5e38\u306b\u7c21\u5358\u3067\u3059\u3002<\/p>\n\n\n\n 2379\u30683355\u306e\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u308b\u3068\u304d<\/p>\n\n\n \u300c700\u3068315\u306e\u6700\u5927\u516c\u7d04\u6570\u300d<\/p>\n\n\n\n \uff1d\u300c315\u306870\u306e\u6700\u5927\u516c\u7d04\u6570\u300d<\/p>\n\n\n\n \uff1d\u300c70\u306835\u306e\u6700\u5927\u516c\u7d04\u6570\u300d<\/p>\n<\/div><\/div>\n\n\n\n \u3068\u3044\u3046\u8003\u3048\u65b9\u3092\u7528\u3044\u3066\u8a08\u7b97\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n \u672c\u8a18\u4e8b\u3067\u306f\u3001\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u3084\u308a\u65b9\u3084\u8a3c\u660e\u306b\u3064\u3044\u3066\u89e3\u8aac<\/span>\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u8a73\u3057\u3044\u624b\u9806\u30841\u6b21\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u306e\u5fdc\u7528\u306b\u3064\u3044\u3066\u3082\u89e3\u8aac\u3057\u3066\u3044\u308b\u306e\u3067\u3001\u305c\u3072\u6700\u5f8c\u307e\u3067\u8aad\u3093\u3067\u304f\u3060\u3055\u3044\u3002<\/span><\/p>\n\n\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306f\u30012\u3064\u306e\u6574\u6570\u306e\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u308b\u8a08\u7b97\u65b9\u6cd5<\/span>\u3067\u3059\u3002<\/p>\n\n\n\n 2\u3064\u4ee5\u4e0a\u306e\u6574\u6570\u306e\u305d\u308c\u305e\u308c\u306e\u7d04\u6570\u306e\u3046\u3061\u3001\u5171\u901a\u306e\u7d04\u6570\u3092\u516c\u7d04\u6570\u3068\u3044\u3044\u307e\u3059\u3002 12\u306e\u7d04\u6570\uff1a1,2,3,4,6,12 12\u306818\u306e\u516c\u7d04\u6570\uff1a1,2,3,6<\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u300112\u306818\u306e\u6700\u5927\u516c\u7d04\u6570\u306f6<\/p>\n<\/div><\/div>\n\n\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u3084\u308a\u65b9\u3092\u89e3\u8aac\u3057\u307e\u3059\u3002<\/span><\/p>\n\n\n\n 315\u3068700\u306e\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u308b\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \u307e\u305a\u306f\u5927\u304d\u3044\u65b9\u306e\u6574\u6570\u3092\u5c0f\u3055\u3044\u6574\u6570\u3067\u5272\u308a\u307e\u3059\u3002<\/p>\n\n\n \u3053\u3053\u3067\u6c42\u3081\u305f\u4f59\u308a\u3092\u4f7f\u3063\u3066\u6b21\u306e\u8a08\u7b97\u3092\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \u5272\u308b\u6570\u3068\u4f59\u308a\u3092\u5de6\u306b\u30ba\u30e9\u3057\u3066\u3001\u3082\u3046\u4e00\u5ea6\u5272\u308a\u7b97\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\n\n\n \u3059\u308b\u3068\u307e\u305f\u4f59\u308a\u304c\u51fa\u3066\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u3053\u308c\u3092\u4f59\u308a\u304c\u51fa\u306a\u304f\u306a\u308b\u307e\u3067\u7e70\u308a\u8fd4\u3057\u307e\u3059\u3002<\/p>\n\n\n 70\u309235\u3067\u5272\u3063\u305f\u3068\u3053\u308d\u3067\u5546\u304c2\u3068\u306a\u308a\u3001\u4f59\u308a\u304c\u7121\u304f\u306a\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001315\u3068700\u306e\u6700\u5927\u516c\u7d04\u6570\u306f35<\/span>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n \u3042\u308c\u3001\u610f\u5916\u3068\u7c21\u5358\u306b\u6700\u5927\u516c\u7d04\u6570\u304c\u6c42\u3081\u3089\u308c\u305f\u3088\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306f\u898b\u305f\u76ee\u304c\u8907\u96d1\u306a\u3060\u3051\u3067\u3001\u8a08\u7b97\u306e\u624b\u9806\u306f\u30b7\u30f3\u30d7\u30eb\u3060\u3088\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306b\u6163\u308c\u308b\u305f\u3081\u306b\u3082\u30461\u554f\u89e3\u3044\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n 3355\u30682379\u306e\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n \u307e\u305a\u306f3355\u30922379\u3067\u5272\u3063\u3066\u4f59\u308a\u3092\u6c42\u3081\u308b\u3053\u3068\u304b\u3089\u59cb\u3081\u307e\u3059\u3002<\/p>\n\n\n \u5272\u308b\u6570\u3092\u4f59\u308a\u3067\u3082\u3046\u4e00\u5ea6\u5272\u308b\u4f5c\u696d\u3092\u7e70\u308a\u8fd4\u3057\u3066\u3044\u304d\u3001\u4f59\u308a\u304c\u51fa\u306a\u304f\u306a\u3063\u305f\u3068\u3053\u308d\u3067\u8a08\u7b97\u7d42\u4e86\u3067\u3059\u3002<\/span><\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u30013355\u30682379\u306e\u6700\u5927\u516c\u7d04\u6570\u306f61<\/span>\u3060\u3068\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n \u6700\u5927\u516c\u7d04\u6570\u3092\u8868\u3059\u8a18\u53f7\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \u300c3355\u30682379\u306e\u6700\u5927\u516c\u7d04\u6570\u300d\u3068\u6bce\u56de\u66f8\u304f\u306e\u306f\u5927\u5909\u306a\u306e\u3067\u3001\u77e5\u3063\u3066\u304a\u304f\u3068\u4fbf\u5229\u3067\u3059\u3002<\/p>\n\n\n\n 3355\u30682379\u306e\u6700\u5927\u516c\u7d04\u6570\u306f61 \u21d4\u3000\\(gcd(3355,2379)=61\\)<\/p>\n<\/div><\/div>\n\n\n\n \u6700\u5927\u516c\u7d04\u6570\u306f\\(gcd(x,y)\\)\u3092\u7528\u3044\u3066\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n 12\u306818\u306a\u3089\u3001\u7d04\u6570\u3092\u3059\u3079\u3066\u66f8\u304d\u51fa\u3057\u3066\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u308b\u3053\u3068\u3082\u53ef\u80fd\u3067\u3059\u3002<\/p>\n\n\n\n \u3057\u304b\u3057\u3001<\/p>\n\n\n\n 31869169\u3068472749749\u306e\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u3088<\/p>\n<\/div><\/div>\n\n\n\n \u3053\u3046\u306a\u308b\u3068\u7d04\u6570\u3092\u3059\u3079\u3066\u66f8\u304d\u51fa\u3059\u306e\u306f\u4e0d\u53ef\u80fd\u306b\u8fd1\u3044\u3067\u3059\u3002<\/span><\/p>\n\n\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306f\u3001\u5927\u304d\u306a\u6570\u306e\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u308b\u3068\u304d\u306b\u4fbf\u5229<\/span>\u3067\u3059\u3002<\/p>\n\n\n\n \u3068\u306f\u3044\u3048\u3001\u5165\u8a66\u554f\u984c\u3067\u3053\u3093\u306a\u306b\u5927\u304d\u306a\u6574\u6570\u306f\u307b\u3068\u3093\u3069\u767b\u5834\u3057\u306a\u3044\u3067\u3059\u3002<\/span><\/p>\n\n\n\n \u5927\u5b66\u5165\u8a66\u306b\u304a\u3044\u3066\u306f\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u308b\u554f\u984c\u3088\u308a\u3082\uff0c\u4e00\u6b21\u4e0d\u5b9a\u65b9\u7a0b\u5f0f ax+by=1 \u306e\u554f\u984c\u3067\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3092\u6d3b\u7528\u3057\u307e\u3059\u3002<\/p>\n\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306f1\u6b21\u4e0d\u5b9a\u5f0f\u306e\u554f\u984c\u3067\u6d3b\u8e8d\u3059\u308b\u3088\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3067\u306f\u3001<\/p>\n\n\n\n \u300c\u5272\u3089\u308c\u308b\u6570\u3068\u5272\u308b\u6570\u306e\u6700\u5927\u516c\u7d04\u6570\u300d\uff1d\u300c\u5272\u308b\u6570\u3068\u3042\u307e\u308a\u306e\u6700\u5927\u516c\u7d04\u6570\u300d<\/span><\/p>\n\n\n\n \u3068\u3044\u3046\u8003\u3048\u65b9\u3092\u7528\u3044\u3066\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n \u3064\u307e\u308a\u4ee5\u4e0b\u306e\u8a08\u7b97\u3067\u306f\u3001<\/p>\n\n\n \u3068\u306a\u308b\u306e\u3067\u3001700\u3068315\u306e\u6700\u5927\u516c\u7d04\u6570\u304c35\u3060\u3068\u3044\u3048\u308b\u306e\u3067\u3059\u3002<\/p>\n\n\n\n \u3053\u3053\u3067\u3001\u306a\u305c\u300c700\u3068315\u306e\u6700\u5927\u516c\u7d04\u6570\u300d\uff1d\u300c70\u306835\u306e\u6700\u5927\u516c\u7d04\u6570\u300d\u304c\u6210\u308a\u7acb\u3064\u306e\u304b<\/span>\u7591\u554f\u3092\u62b1\u304f\u5b66\u751f\u3082\u591a\u3044\u3067\u3057\u3087\u3046\u3002<\/p>\n\n\n\n \u305d\u3093\u306a\u3042\u306a\u305f\u306b\u5411\u3051\u3066\u3001\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u8a3c\u660e\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \u307e\u305a\u306f\u4ee5\u4e0b\u306e\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n 2\u3064\u306e\u6574\u6570\\(a,b(a\u2265b>0)\\)\u306b\u3064\u3044\u3066a\u3092b\u3067\u5272\u3063\u305f\u4f59\u308a\u3092r\u3068\u3059\u308b\u3068\u304d\u3001a\u3068b\u306e\u6700\u5927\u516c\u7d04\u6570\u306fb\u3068r\u306e\u6700\u5927\u516c\u7d04\u6570\u3068\u7b49\u3057\u3044<\/p>\n\n\n\n \u300a\u8a3c\u660e\u300b<\/span><\/p>\n\n\n\n a\u3068b\u306e\u6700\u5927\u516c\u7d04\u6570\u3092G\u3068\u3059\u308b\u3068\u3001 \\(a=bQ+r \\cdots \u2461\\)<\/p>\n\n\n\n \u2461\u306b\u2460\u3092\u4ee3\u5165\u3057\u3066\u6574\u7406\u3059\u308b\u3068 \\(a’-Qb’=r’\\)\u3068\u7f6e\u304f\u3068 \u3088\u3063\u3066b\u3068r\u306fG\u3092\u516c\u7d04\u6570\u306b\u6301\u3064\u3002 \u3053\u308c\u3067\u8a3c\u660e\u7d42\u4e86\u3067\u3059\u3002<\/p>\n\n\n\n \u3053\u308c\u306f<\/p>\n\n\n\n \u300c\u5272\u3089\u308c\u308b\u6570\u3068\u5272\u308b\u6570\u306e\u6700\u5927\u516c\u7d04\u6570\u300d\uff1d\u300c\u5272\u308b\u6570\u3068\u3042\u307e\u308a\u306e\u6700\u5927\u516c\u7d04\u6570\u300d<\/span><\/p>\n\n\n\n \u3092\u610f\u5473\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \u3064\u307e\u308a\u3001\u300c\u5272\u308a\u5207\u308c\u308b\u307e\u3067\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3092\u7d9a\u3051\u305f\u3068\u304d\u306e\u6700\u5f8c\u306e\u5272\u308b\u6570\u304c\u6700\u5927\u516c\u7d04\u6570\u3067\u3042\u308b\u300d<\/span>\u3068\u3044\u3046\u3053\u3068\u304c\u8a00\u3048\u308b\u306e\u3067\u3059\u3002<\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n \u300c700\u3068315\u306e\u6700\u5927\u516c\u7d04\u6570\u300d \u3068\u306a\u308a\u3001700\u3068315\u306e\u6700\u5927\u516c\u7d04\u6570\u304c35\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \u3053\u3053\u307e\u3067\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3092\u89e3\u8aac\u3057\u307e\u3057\u305f\u304c\u3001\u5927\u5b66\u5165\u8a66\u3067\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u308b\u3060\u3051\u306e\u554f\u984c\u306f\u307b\u3068\u3093\u3069\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n\n\n\n \u5165\u8a66\u306b\u304a\u3044\u3066\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u304c\u4e00\u756a\u6d3b\u8e8d\u3059\u308b\u306e\u306f\u300c1\u6b21\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u300d\u306e\u554f\u984c\u3078\u306e\u5229\u7528\u3067\u3059\u3002<\/p>\n\n\n\n 1\u6b21\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u3067\u306f\u6b21\u306e\u3088\u3046\u306a\u554f\u984c\u304c\u51fa\u307e\u3059\u3002<\/p>\n\n\n\n \\(2x+3y=1\\)\u3092\u6e80\u305f\u3059\u6574\u6570\u89e3\u3092\u3059\u3079\u3066\u6c42\u3081\u3088\u3002<\/p>\n<\/div><\/div>\n\n\n\n \u307e\u305a\u306f\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\u6574\u6570\u89e3\u30921\u3064\u898b\u3064\u3051\u307e\u3059\u3002<\/p>\n\n\n\n \u3053\u308c\u304f\u3089\u3044\u5c0f\u3055\u3044\u6570\u5b57\u306a\u3089\u3070\u3001\u5b9f\u969b\u306b\u6570\u5b57\u3092\u4ee3\u5165\u3057\u3066\u6574\u6570\u89e3\u3092\u898b\u3064\u3051\u307e\u3059\u3002<\/p>\n\n\n\n \\(2x-3y=1 \\cdots \u2460\\)<\/p>\n\n\n\n \u3053\u306e\u65b9\u7a0b\u5f0f\u306f\\((x,y)=(2,1)\\)\u3067\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \\(2 \\cdot 2 -3 \\cdot 1=1 \\cdots \u2461\\)<\/p>\n\n\n\n \\(\u2460-\u2461\\)<\/p>\n\n\n\n \\(2(x-2)-3(y-1)=0\\)<\/p>\n\n\n\n \u3053\u308c\u3092\u5909\u5f62\u3059\u308b\u3068<\/p>\n\n\n\n \\(2(x-2)=3(y-1) \\cdots \u2462\\)<\/p>\n\n\n\n\n\n\n\n 2\u30683\u306f\u4e92\u3044\u306b\u7d20\u306a\u306e\u3067\u3001\\(x-2\\)\u306f3\u306e\u500d\u6570\u3060\u3068\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n \u3088\u3063\u3066\u3001k\u3092\u6574\u6570\u3068\u3057\u3066\u3001\\(x\u22122=3k\\)<\/span>\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u3053\u308c\u3092\u2462\u306b\u4ee3\u5165\u3059\u308b\u3068<\/p>\n\n\n\n \\(2\u22c53k=3(y-1)\\)<\/p>\n\n\n\n \\(\u2234\u3000y-1=2k\\)<\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001\u6c42\u3081\u308b\u89e3\u306f<\/p>\n\n\n\n \\(x=3k+2, y=2k+1\uff08k\u306f\u6574\u6570\uff09\\)\u22ef\u3010\u7b54\u3011<\/p>\n\n\n\n 1\u6b21\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u306e\u89e3\u304d\u65b9\u3092\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u6b21\u306e\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\u6574\u6570\u89e3\u3092\u3059\u3079\u3066\u6c42\u3081\u3088\u3002<\/p>\n\n\n\n \\(275x+61y=1\\)<\/p>\n<\/div><\/div>\n\n\n\n \u3053\u3053\u3067\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u304c\u975e\u5e38\u306b\u6d3b\u8e8d\u3057\u307e\u3059\u3002<\/span><\/p>\n\n\n\n \u307e\u305a\u306f275\u306861\u3067\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n \u4f59\u308a\u304c1\u306b\u306a\u308b\u307e\u3067\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3092\u7e70\u308a\u8fd4\u3057\u307e\u3059\u3002<\/span><\/p>\n\n\n\n \u3053\u3053\u304b\u3089\u306f\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u308b\u3068\u304d\u3068\u306f\u7570\u306a\u308b\u624b\u9806\u306a\u306e\u3067\u6ce8\u76ee\u3067\u3059\u3002<\/p>\n\n\n \u4e00\u756a\u4e0b\u306e\u5f0f\u306b\u3001\u4e0a\u306e\u5f0f\u3092\u9806\u306b\u4ee3\u5165\u3057\u3066\u3044\u304f\u3068<\/p>\n\n\n\n \\begin{aligned} \u3086\u3048\u306b\u3001 \u3057\u305f\u304c\u3063\u3066\u3001275x+61y=1\u306e\u6574\u6570\u89e3\u306e1\u3064\u306f x=2, y=\u22129 \u3060\u3068\u308f\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u2460\uff0d\u2461\u3088\u308a<\/p>\n\n\n\n \\(275(x\u22122)+61(y+9)=0\\)<\/p>\n\n\n\n \\(\u2234\u3000275(x\u22122)=\u221261(y+9) \\cdots \u2462\\)<\/p>\n\n\n\n 275\u306861\u306f\u4e92\u3044\u306b\u7d20\u3060\u304b\u3089\u3001\\(x\u22122\\)\u306f61\u306e\u500d\u6570\u3068\u3044\u3048\u307e\u3059\u3002<\/p>\n\n\n\n \u3088\u3063\u3066\u3001k\u3092\u6574\u6570\u3068\u3057\u3066\u3001\\(x\u22122=61k\\)\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u3053\u308c\u3092\u2462\u306b\u4ee3\u5165\u3059\u308b\u3068<\/p>\n\n\n\n \\(275\u22c561k=\u221261(y+9)\\)<\/p>\n\n\n\n \\(\u2234\u3000y+9=\u2212275k\\)<\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001\u6c42\u3081\u308b\u89e3\u306f<\/p>\n\n\n\n \\(x=61k+2, y=\u2212275k\u22129\uff08k\u306f\u6574\u6570)\\)<\/p>\n\n\n\n \u3053\u3053\u307e\u3067\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3092\u6d3b\u7528\u3057\u3066\u554f\u984c\u3092\u89e3\u304f\u7df4\u7fd2\u3092\u3057\u307e\u3059\u3002<\/p>\n\n\n\n 3465\u30687812\u306e\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u3088\u3002<\/p>\n<\/div><\/div>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u30013465\u30687812\u306e\u6700\u5927\u516c\u7d04\u6570\u306f63\u3067\u3059\u3002<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n \u6b21\u306b1\u6b21\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u306e\u6574\u6570\u89e3\u3092\u6c42\u3081\u308b\u554f\u984c\u3082\u7df4\u7fd2\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n \u6b21\u306e\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\u6574\u6570\u89e3\u3092\u3059\u3079\u3066\u6c42\u3081\u3088\u3002<\/p>\n\n\n\n \\(92x+197y=1\\)<\/p>\n<\/div><\/div>\n\n\n\n \u307e\u305a\u306f\\(92x+197y=1\\)\u3092\u6e80\u305f\u3059\u6574\u6570\u89e3\u306e1\u3064\u3092\u898b\u3064\u3051\u307e\u3059\u3002<\/p>\n\n\n\n \\(92x+197y=1 \\cdots\u3000\u2460\\)<\/p>\n\n\n\n \u3068\u3059\u308b\u3002<\/p>\n\n\n\n \u3053\u308c\u3089\u3092\u5f0f\u5909\u5f62\u3057\u3066\u3044\u304f\u3068<\/p>\n\n\n\n \\(1 =92-13 \\cdot 7\\) \u3057\u305f\u304c\u3063\u3066\u3001\\((x,y)=(-13,-7)\\)\u306f\u2460\u3092\u6e80\u305f\u30591\u3064\u306e\u6574\u6570\u89e3\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \\(92x+197y=1 \\cdots\u3000\u2460\\) \u2460-\u2461\u3088\u308a<\/p>\n\n\n\n \\(92(x+13)+197(y+7)=0\\)<\/p>\n\n\n\n \\(92(x+13)=-197(y+7) \\cdots \u2462\\)<\/p>\n\n\n\n \u3053\u3053\u306792\u3068197\u306f\u4e92\u3044\u306b\u7d20\u306a\u306e\u3067\u3001x+13\u304c197\u306e\u500d\u6570\u3067\u3042\u308b\u3002<\/p>\n\n\n\n \u3088\u3063\u3066\u3001k\u3092\u6574\u6570\u3068\u3057\u3066\u3001\\(x+13=197k\\)\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n \u3053\u308c\u3092\u2462\u306b\u4ee3\u5165\u3059\u308b\u3068<\/p>\n\n\n\n \\(92\u22c5197k=\u2212197(y+7)\\)<\/p>\n\n\n\n \\(\u2234\u3000y+7=\u221292k\\)<\/p>\n\n\n\n \u3057\u305f\u304c\u3063\u3066\u3001\u6c42\u3081\u308b\u89e3\u306f<\/p>\n\n\n\n \\(x=197k-13, y=\u221292k\u22127\uff08k\u306f\u6574\u6570\uff09\\)<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n \u4eca\u56de\u306f\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3057\u305f\u3002<\/p>\n\n\n\n \u30fb\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306f\uff12\u3064\u306e\u6574\u6570\u306e\u6700\u5927\u516c\u7d04\u6570\u3092\u6c42\u3081\u308b\u8a08\u7b97<\/p>\n\n\n\n \u30fb\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u3084\u308a\u65b9<\/p>\n\n\n\n \u30fb\u4e00\u6b21\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u3067\u6d3b\u8e8d<\/p>\n\n\n\n
\u300c1\u6b21\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u304c\u82e6\u624b\u300d<\/span>
\u4eca\u56de\u306f\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306b\u95a2\u3059\u308b\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002<\/p>\n\n\n
\u9ad8\u6821\u751f<\/span><\/div>
<\/figure>\n<\/div>\n\n\n
\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3067\u306f<\/p>\n\n\n\n\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3068\u306f\uff1f<\/h2>\n\n\n
<\/figure>\n<\/div>\n\n\n
\u516c\u7d04\u6570\u306e\u4e2d\u3067\u3082\u6700\u5927\u306e\u516c\u7d04\u6570\u3092\u3001\u6700\u5927\u516c\u7d04\u6570\u3068\u3044\u3044\u307e\u3059\u3002<\/p>\n\n\n\n
18\u306e\u7d04\u6570\uff1a1,2,3,6,9,12<\/p>\n\n\n\n\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u3084\u308a\u65b9<\/h2>\n\n\n\n
<\/figure>\n<\/div>\n\n\n
<\/figure>\n<\/div>\n\n\n
<\/figure>\n<\/div>\n\n\n
\u9ad8\u6821\u751f<\/span><\/div>
\u30b7\u30fc\u30bf<\/span><\/div>\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u5177\u4f53\u4f8b<\/h2>\n\n\n\n
<\/figure>\n<\/div>\n\n\n
<\/figure>\n<\/div>\n\n\n\u6700\u5927\u516c\u7d04\u6570\u306e\u8a18\u53f7<\/h2>\n\n\n\n
\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u30e1\u30ea\u30c3\u30c8<\/h2>\n\n\n\n
\u30b7\u30fc\u30bf<\/span><\/div>\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u306e\u8a3c\u660e<\/h2>\n\n\n\n
<\/figure>\n<\/div>\n\n\n
\u300c700\u3068315\u306e\u6700\u5927\u516c\u7d04\u6570\u300d
\uff1d\u300c315\u306870\u306e\u6700\u5927\u516c\u7d04\u6570\u300d
\uff1d\u300c70\u306835\u306e\u6700\u5927\u516c\u7d04\u6570\u300d<\/p>\n\n\n\n
\\(a=Ga’,b=Gb’ (a’\u3068b’\u306f\u4e92\u3044\u306b\u7d20) \\cdots \u2460\\)
\u3068\u66f8\u3051\u308b\u3002<\/p>\n\n\n\n
\\(G(a’-Qb’)=r\\)<\/p>\n\n\n\n
\\(r=Gr’\\)<\/p>\n\n\n\n
\u307e\u305f\u3001\u2461\u3088\u308ab\u3068r\u306fG\u3088\u308a\u5927\u304d\u306a\u516c\u7d04\u6570\u3092\u6301\u305f\u306a\u3044\u306e\u3067b\u3068r\u306e\u6700\u5927\u516c\u7d04\u6570\u306fG\u3067\u3042\u308b\u3002<\/p>\n\n\n\n
2\u3064\u306e\u6574\u6570\\(a,b(a\u2265b>0)\\)\u306b\u3064\u3044\u3066a\u3092b\u3067\u5272\u3063\u305f\u4f59\u308a\u3092r\u3068\u3059\u308b\u3068\u304d\u3001a\u3068b\u306e\u6700\u5927\u516c\u7d04\u6570\u306fb\u3068r\u306e\u6700\u5927\u516c\u7d04\u6570\u3068\u7b49\u3057\u3044<\/p>\n\n\n\n
\uff1d\u300c315\u306870\u306e\u6700\u5927\u516c\u7d04\u6570\u300d
\uff1d\u300c70\u306835\u306e\u6700\u5927\u516c\u7d04\u6570\u300d<\/p>\n\n\n\n\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3068\u4e0d\u5b9a\u65b9\u7a0b\u5f0f<\/h2>\n\n\n\n
\u4e00\u6b21\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u306e\u57fa\u790e\u3092\u78ba\u8a8d<\/h3>\n\n\n\n
\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u3092\u7528\u3044\u3066\u4e0d\u5b9a\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f<\/h3>\n\n\n\n
<\/figure>\n<\/div>\n\n\n
<\/figure>\n<\/div>\n\n\n
1 &=31-30 \\cdot 1 \\\\
&=31-(61-31 \\cdot 1) \\cdot 1 \\\\
&=31 \\cdot 2+61 \\cdot(-1) \\\\
&=(275-61 \\cdot 4) \\cdot 2+61 \\cdot(-1) \\\\
&=275 \\cdot 2+61 \\cdot(-9)
\\end{aligned}<\/p>\n\n\n\n
\\(275 \\cdot 2+61 \\cdot(-9)=1 \\cdots \u2461\\)<\/p>\n\n\n\n\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u300a\u7df4\u7fd2\u554f\u984c\u300b<\/h2>\n\n\n\n
\u89e3\u7b54\u306f\u3053\u3061\u3089<\/span><\/i><\/i><\/span><\/summary>
<\/figure>\n<\/div>\n\n\n\u89e3\u7b54\u306f\u3053\u3053\u3092\u30af\u30ea\u30c3\u30af<\/span><\/i><\/i><\/span><\/summary>
\\(=92-(197-92 \\cdot 2) \\cdot 7\\)
\\(=92 \\cdot (-13)-197 \\cdot 7\\)
\\(=92 \\cdot (-13)+197 \\cdot(-7) \\cdots \u2461\\)<\/p>\n\n\n\n
\\(92 \\cdot (-13)+197 \\cdot(-7)=1 \\cdots \u2461\\)<\/p>\n\n\n\n\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u306e\u4e92\u9664\u6cd5\u3000\u307e\u3068\u3081<\/h2>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n