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銆岃緱杞浉闄ゆ硶銆嶅張鍙仛銆屾鍑犻噷寰楃畻娉曘?鏄叕鍏冨墠 300 騫村乏鍙崇殑甯岃厞鏁板瀹舵鍑犻噷寰楀湪浠栫殑钁椾綔銆婂嚑浣曞師鏈嬫彁鍑虹殑.鍒╃敤榪欎釜鏂規(guī)硶,鍙互杈冨揩鍦版眰鍑轟袱涓嚜鐒舵暟鐨勬渶澶у叕鍥犳暟,鍗?HCF 鎴栧彨鍋?gcd.
鏈澶у叕綰︽暟錛坓reatest common divisor錛岀畝鍐欎負(fù)gcd錛涙垨highest common factor錛岀畝鍐欎負(fù)hcf)
鎵璋撴渶澶у叕鍥犳暟,鏄寚鍑犱釜鏁扮殑鍏辨湁鐨勫洜鏁頒箣涓渶澶х殑涓涓?渚嬪 8 鍜?12 鐨勬渶澶у叕鍥犳暟鏄?4,璁頒綔 gcd(8,12)=4.
鍦ㄤ粙緇嶈繖涓柟娉曚箣鍓?鍏堣鏄庢暣闄ゆх殑涓浜涚壒鐐?娉ㄤ互涓嬫枃鐨勬墍鏈夋暟閮芥槸姝ｆ暣鏁?浠ュ悗涓嶅啀閲嶈.
鎴戜滑鍙互榪欐牱緇欏嚭鏁撮櫎浠ョ殑瀹氫箟:
瀵規(guī)柤涓や釜鑷劧鏁?a 鍜?b,鑻ュ瓨鍦ㄦ鏁存暟 q,浣垮緱 a=bq,鍒?b 鑳芥暣闄?a,璁頒綔 b | a,鎴戜滑鍙?b 鏄?a 鐨勫洜鏁?鑰?a 鏄?b 鐨勫嶆暟.
閭ｉ杭濡傛灉 c | a,鑰屼笖 c | b,鍒?c 鏄?a 鍜?b 鐨勫叕鍥犳暟.
鐢辨,鎴戜滑鍙互寰楀嚭浠ヤ笅涓浜涙帹璁?
鎺ㄨ涓:濡傛灉 a | b,鑻?k 鏄暣鏁?鍒?a | kb.鍥犱負(fù)鐢?a | b 鍙煡 ha=b,鎵浠?(hk)a=kb,鍗?a | kb.
鎺ㄨ浜?濡傛灉 a | b 浠ュ強(qiáng) a | c,鍒?a | (b±c).鍥犱負(fù)鐢?a | b 浠ュ強(qiáng) a | c,鍙煡 ha=b,ka=c,浜屽紡鐩稿姞,寰?(h+k)a=b+c,鍗?a | (b+c).鍚屾牱鎶婁簩寮忕浉鍑忓彲寰?a | (b-c).
鎺ㄨ涓?濡傛灉 a | b 浠ュ強(qiáng) b | a,鍒?a=b.鍥犱負(fù)鐢?a | b 浠ュ強(qiáng) b | a,鍙煡 ha=b,a=kb,鍥犳 a=k(ha),hk=1,鐢辨柤 h 鍜?k 閮芥槸姝ｆ暣鏁?鏁?h=k=1,鍥犳 a=b.
杈楄漿鐩擱櫎娉曟槸鐢ㄦ潵璁＄畻涓や釜鏁扮殑鏈澶у叕鍥犳暟,鍦ㄦ暟鍊煎緢澶ф椂灝ゅ叾鏈夌敤鑰屼笖搴旂敤鍦ㄧ數(shù)鑴戠▼寮忎笂涔熷崄鍒嗙畝鍗?鍏剁悊璁哄涓?
濡傛灉 q 鍜?r 鏄?m 闄や互 n 鐨勫晢鍙?qiáng)浣欐?鍗?m=nq+r,鍒?gcd(m,n)=gcd(n,r).
璇佹槑鏄繖鏍風(fēng)殑:
璁?a=gcd(m,n),b=gcd(n,r)
鍒欐湁 a | m 鍙?a | n,鍥犳 a | (m-nq)(榪欐槸鐢辨帹璁轟竴鍙?qiáng)鎺ㄨZ簩寰楀嚭鐨?,鍗?a | r 鍙?a | n,鎵浠?a | b
鍙?b | r 鍙?b | n,鎵浠?b | (nq+r),鍗?b | m 鍙?b | n,鎵浠 | a.鍥犱負(fù) a | b 騫朵笖 b | a,鎵浠?a=b,鍗?gcd(m,n)=gcd(n,r).
渚嬪璁＄畻 gcd(546, 429),鐢辨柤 546=1(429)+117,429=3(117)+78,117=1(78)+39,78=2(39),鍥犳
gcd(546, 429)
=gcd(429, 117)
=gcd(117, 78)
=gcd(78, 39)
=39
Java瀹炵幇浠ｇ爜濡備笅:
package com;public class GcdTest
{
//寰幆瀹炵幇 int gcd1(int a, int b)
{
int k = 0; do { //寰楀埌浣欐暟 k = a % b; //鏍規(guī)嵁杈楄漿鐩擱櫎娉?鎶婅闄ゆ暟璧嬬粰闄ゆ暟 a = b; //浣欐暟璧嬬粰琚櫎鏁?/span> b = k;
} while (k != 0); //榪斿洖琚櫎鏁?/span> return a; } //閫嗗綊瀹炵幇 int gcd2(int a,int b) { //鐩村埌婊¤凍姝ゆ潯浠墮嗗綊閫鍑?/span> if(b == 0) { return a; } if(a < 0) { return gcd2(-a,b); } if(b < 0) { return gcd2(a,-b); } return gcd2(b,a % b); } public static void main(String[] args) { GcdTest gt = new GcdTest(); System.out.println(gt.gcd1(888,458)); System.out.println(gt.gcd2(888, 458)); }
}]]>