시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 (추가 시간 없음) 256 MB 35 9 8 23.529%

문제

당신은 N개의 작은 섬으로 이루어진 다도해에 다리를 건설하여 임의의 한 섬에서 임의의 다른 섬으로 도달 가능하게 하고 싶다. 현재는 아무 다리도 없는 상태이다.

섬은 0번 부터 N-1번까지 번호가 붙어있다.

당신은 아래와 같은 알고리즘에 따라 다리를 건설하기로 했다.

  • 세 개의 양의 정수 Seed, A, B 를 고른다. 이후, E[ i ], X[ i ], Y[ i ] 를 아래와 같이 정의한다.
    • E[ 1 ] = Seed % N 그리고 E[ i ] = (E[ i-1 ] * A + B) % N2 (i > 1). (각각의 E[i] 값은 어느 두 섬을 연결하는 다리를 지을지 결정한다.)
    • X[ i ] = E[ i ] / N 그리고 Y[ i ] = E[ i ] % N (i ≥ 1).
    • 위 두 수식에서 "%" 는 정수 나머지 연산을 나타내는 Modulo 이며, "/"은 정수 나눗셈 연산이다.
  • 오늘부터 1일 후 (즉, 내일) X[ 1 ] 번 섬과 Y[ 1 ]번 섬을 연결하는 다리를 건설한다. 만약 X[ 1 ] = Y[ 1 ]이면 이 날은 건설은 하지 않고 논다.
  • 오늘 부터 i일 후, X[ i ] 번 섬과 Y[ i ] 번 섬을 연결하는 다리를 건설한다. 만약 X[ i ] = Y[ i ] 이거나 이미 두 섬을 잇는 다리가 있다면, 이 날은 건설은 하지 않고 논다.

위 알고리즘에 따라 다리를 건설 할 때, 당신은 오늘 부터 몇 일이 지난 후 (M일 후) 목적을 달성할지 궁금하다.

예를 들어 N = 4, Seed = 2020, A = 2, B = 3 인 경우를 생각해보자. 

  • E[ 1 ] = 2020 % 16 = 4, X[ 1 ] = 4 / 4 = 1, Y[ 1 ] = 4 % 4 = 0. 따라서 (1번섬, 0번섬) 을 잇는 다리를 건설한다.
  • E[ 2 ] = (4 * 2 + 3) % 16 = 11, X[ 2 ] = 11 / 4 = 2, Y[ 2 ] = 11 % 4 = 3. 따라서 (2번섬, 3번섬)을 잇는 다리를 건설한다.
  • E[ 3 ] = (11 * 2 + 3) % 16 = 9, X[ 3 ] = 9 / 4 = 2, Y[ 3 ] = 9 % 4 = 1. 따라서 (2번섬, 1번섬)을 잇는 다리를 건설한다.
  • 세 번째 다리를 건설한 후 임의의 한 섬에서 다른 섬으로 도달할 수 있으므로 이 때 M = 3이다.

다른 예로, N = 4, Seed = 2020, A = 3, B = 4 인 경우를 생각해보자.

  • E[ 1 ] = 2020 % 16 = 4, X[ 1 ] = 4 / 4 = 1, Y[ 1 ] = 4 % 4 = 0. 따라서 (1번섬, 0번섬) 을 잇는 다리를 건설한다.
  • E[ 2 ] = (4 * 3 + 4) % 16 = 0, X[ 2 ] = 0 / 4 = 0, Y[ 2 ] = 0 % 4 = 0. X,Y 값이 같으므로 다리를 건설하지 않는다.
  • E[ 3 ] = (0 * 3 + 4) % 16 = 4, X[ 3 ] = 4 / 4 = 1, Y[ 3 ] = 4 % 4 = 0. 이미 다리가 있으므로 다리를 건설하지 않는다.
  • 이 경우, (1번섬, 0번섬)을 잇는 다리 이외에 다른 다리를 건설하지 못한다. 따라서 이 때 위 조건을 만족하는 M은 존재하지 않는다.

입력으로 N, Seed, A, B가 주어졌을 때, 위 조건을 만족하는 M값을 구하여 출력한다. 만약 조건을 만족하는 M값이 없다면 0을 출력한다.

입력

첫 줄에 테스트 케이스의 수 T가 주어진다.

각 테스트 케이스는 네 개의 정수가 공백으로 구분되어 주어지는데, 순서대로 N, Seed, A, B 이다.

출력

조건을 만족하는 M이 존재하면 그 중 최소값을 출력하고, 존재하지 않으면 0을 출력한다.

제한

  • 1 ≤ T ≤ 10
  • 2 ≤ N ≤ 1,000
  • 1 ≤ Seed, A, B ≤ 1,000,000,000

예제 입력 1

4
4 2020 2 3
4 2020 3 4
4 2020 9 7
5 2020 4 7

예제 출력 1

3
0
6
5
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(\uac01\uac01\uc758 E[i] \uac12\uc740 \uc5b4\ub290 \ub450 \uc12c\uc744 \uc5f0\uacb0\ud558\ub294 \ub2e4\ub9ac\ub97c \uc9c0\uc744\uc9c0 \uacb0\uc815\ud55c\ub2e4.)<\/li>\r\n\t\t<li>X[ i ] = E[ i ] \/ N&nbsp;\uadf8\ub9ac\uace0 Y[ i ] = E[ i ] % N (i &ge; 1).<\/li>\r\n\t\t<li>\uc704 \ub450 \uc218\uc2dd\uc5d0\uc11c&nbsp;&quot;%&quot; \ub294 \uc815\uc218 \ub098\uba38\uc9c0 \uc5f0\uc0b0\uc744 \ub098\ud0c0\ub0b4\ub294 Modulo \uc774\uba70, &quot;\/&quot;\uc740 \uc815\uc218 \ub098\ub217\uc148 \uc5f0\uc0b0\uc774\ub2e4.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n\t<li>\uc624\ub298\ubd80\ud130 1\uc77c \ud6c4 (\uc989, \ub0b4\uc77c)&nbsp;X[ 1&nbsp;] \ubc88 \uc12c\uacfc Y[ 1&nbsp;]\ubc88 \uc12c\uc744 \uc5f0\uacb0\ud558\ub294 \ub2e4\ub9ac\ub97c \uac74\uc124\ud55c\ub2e4. \ub9cc\uc57d X[ 1&nbsp;] = Y[ 1&nbsp;]\uc774\uba74 \uc774 \ub0a0\uc740 \uac74\uc124\uc740 \ud558\uc9c0 \uc54a\uace0 \ub17c\ub2e4.<\/li>\r\n\t<li>\uc624\ub298 \ubd80\ud130 i\uc77c \ud6c4, X[ i&nbsp;] \ubc88 \uc12c\uacfc Y[ i&nbsp;] \ubc88 \uc12c\uc744 \uc5f0\uacb0\ud558\ub294 \ub2e4\ub9ac\ub97c \uac74\uc124\ud55c\ub2e4. \ub9cc\uc57d X[ i ] = Y[ i ] \uc774\uac70\ub098 \uc774\ubbf8 \ub450 \uc12c\uc744 \uc787\ub294 \ub2e4\ub9ac\uac00 \uc788\ub2e4\uba74, \uc774 \ub0a0\uc740 \uac74\uc124\uc740 \ud558\uc9c0 \uc54a\uace0 \ub17c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc704 \uc54c\uace0\ub9ac\uc998\uc5d0 \ub530\ub77c \ub2e4\ub9ac\ub97c \uac74\uc124 \ud560 \ub54c, \ub2f9\uc2e0\uc740 \uc624\ub298 \ubd80\ud130 \uba87 \uc77c\uc774 \uc9c0\ub09c \ud6c4 (M\uc77c \ud6c4) \ubaa9\uc801\uc744 \ub2ec\uc131\ud560\uc9c0 \uad81\uae08\ud558\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4 N = 4, Seed = 2020, A = 2, B = 3 \uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud574\ubcf4\uc790.&nbsp;<\/p>\r\n\r\n<ul>\r\n\t<li>E[ 1 ] = 2020 % 16 = 4, X[ 1 ] = 4 \/ 4 = 1, Y[ 1 ] = 4 % 4 = 0. \ub530\ub77c\uc11c (1\ubc88\uc12c, 0\ubc88\uc12c) \uc744 \uc787\ub294 \ub2e4\ub9ac\ub97c \uac74\uc124\ud55c\ub2e4.<\/li>\r\n\t<li>E[ 2 ] = (4 * 2 + 3) % 16 = 11, X[ 2 ] = 11 \/ 4 = 2, Y[ 2 ] = 11 % 4 = 3. \ub530\ub77c\uc11c (2\ubc88\uc12c, 3\ubc88\uc12c)\uc744 \uc787\ub294 \ub2e4\ub9ac\ub97c \uac74\uc124\ud55c\ub2e4.<\/li>\r\n\t<li>E[ 3 ] = (11 * 2 + 3) % 16 = 9, X[ 3 ] = 9 \/ 4 = 2, Y[ 3 ] = 9 % 4 = 1. \ub530\ub77c\uc11c (2\ubc88\uc12c, 1\ubc88\uc12c)\uc744 \uc787\ub294 \ub2e4\ub9ac\ub97c \uac74\uc124\ud55c\ub2e4.<\/li>\r\n\t<li>\uc138 \ubc88\uc9f8 \ub2e4\ub9ac\ub97c \uac74\uc124\ud55c \ud6c4 \uc784\uc758\uc758 \ud55c \uc12c\uc5d0\uc11c \ub2e4\ub978 \uc12c\uc73c\ub85c \ub3c4\ub2ec\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c \uc774 \ub54c M = 3\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\ub2e4\ub978 \uc608\ub85c, N = 4, Seed = 2020, A = 3, B = 4 \uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud574\ubcf4\uc790.<\/p>\r\n\r\n<ul>\r\n\t<li>E[ 1 ] = 2020 % 16 = 4, X[ 1 ] = 4 \/ 4 = 1, Y[ 1 ] = 4 % 4 = 0. \ub530\ub77c\uc11c (1\ubc88\uc12c, 0\ubc88\uc12c) \uc744 \uc787\ub294 \ub2e4\ub9ac\ub97c \uac74\uc124\ud55c\ub2e4.<\/li>\r\n\t<li>E[ 2 ] = (4 * 3&nbsp;+ 4) % 16 = 0, X[ 2 ] = 0&nbsp;\/ 4&nbsp;= 0, Y[ 2 ] = 0 % 4 = 0.&nbsp;X,Y&nbsp;\uac12\uc774 \uac19\uc73c\ubbc0\ub85c \ub2e4\ub9ac\ub97c \uac74\uc124\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\r\n\t<li>E[ 3 ] = (0&nbsp;* 3&nbsp;+ 4) % 16 = 4, X[ 3 ] = 4 \/ 4 = 1, Y[ 3 ] = 4 % 4 = 0. \uc774\ubbf8 \ub2e4\ub9ac\uac00 \uc788\uc73c\ubbc0\ub85c \ub2e4\ub9ac\ub97c \uac74\uc124\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\r\n\t<li>\uc774 \uacbd\uc6b0, (1\ubc88\uc12c, 0\ubc88\uc12c)\uc744 \uc787\ub294 \ub2e4\ub9ac \uc774\uc678\uc5d0 \ub2e4\ub978 \ub2e4\ub9ac\ub97c \uac74\uc124\ud558\uc9c0 \ubabb\ud55c\ub2e4. \ub530\ub77c\uc11c \uc774 \ub54c \uc704 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 M\uc740 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc785\ub825\uc73c\ub85c N, Seed, A, B\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uc704 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 M\uac12\uc744 \uad6c\ud558\uc5ec \ucd9c\ub825\ud55c\ub2e4. \ub9cc\uc57d \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 M\uac12\uc774 \uc5c6\ub2e4\uba74 0\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","input":"<p>\uccab \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc218 T\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ub124 \uac1c\uc758 \uc815\uc218\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c0\ub294\ub370, \uc21c\uc11c\ub300\ub85c N, Seed, A, B \uc774\ub2e4.<\/p>\r\n","output":"<p>\uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 M\uc774 \uc874\uc7ac\ud558\uba74 \uadf8 \uc911 \ucd5c\uc18c\uac12\uc744 \ucd9c\ub825\ud558\uace0, \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\uba74 0\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean","limit":"<ul>\r\n\t<li>1 &le; T &le; 10<\/li>\r\n\t<li>2 &le; N &le; 1,000<\/li>\r\n\t<li>1 &le; Seed, A, B &le; 1,000,000,000<\/li>\r\n<\/ul>\r\n"},{"problem_id":"19593","problem_lang":"1","title":"Islands","description":"<p>You are trying to build bridges to connect N islands such that you can reach any island from any island. Currently, there are no bridges.<\/p>\r\n\r\n<p>Islands are labeled as 0 to N-1, inclusive.<\/p>\r\n\r\n<p>You&#39;ve decided to build the bridges according to the following procedure:<\/p>\r\n\r\n<ul>\r\n\t<li>First, you pick three positive integers, Seed, A, and B. Then, you define E[ i ], X[ i ], and Y[ i ] as follows.\r\n\t<ul>\r\n\t\t<li>E[ 1 ] = Seed % N<sup>2&nbsp;<\/sup>&nbsp;and E[ i ] = (E[ i-1 ] * A + B) % N<sup>2<\/sup> (i &gt;&nbsp;1). (Each E[i] describes which two islands you&#39;ll connect by building a bridge.)<\/li>\r\n\t\t<li>X[ i ] = E[ i ] \/ N&nbsp;and Y[ i ] = E[ i ] % N (i &ge; 1).<\/li>\r\n\t\t<li>In the definition above, &quot;%&quot; is the modulo operation and &quot;\/&quot; is the integer division operation.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n\t<li>In 1 day from today (i.e., tomorrow), you&#39;ll build a bridge that would connect X[ 1 ] and Y[ 1 ]. If X[ 1 ] = Y[ 1 ], then you do not build any bridges on this day.<\/li>\r\n\t<li>In i days from today, you&#39;ll build a bridge that would connect X[ i ] and Y[ i ]. If X[ i ] = Y[ i ] or if there is already a bridge connecting these two islands, then you do not build any bridges on this day.&nbsp;<\/li>\r\n<\/ul>\r\n\r\n<p>You&#39;re curious to know in how many days (in M days), you&#39;ll be able to accomplish your goal.&nbsp;<\/p>\r\n\r\n<p>For instance, suppose that&nbsp;N = 4, Seed = 2020, A = 2, and B = 3.<\/p>\r\n\r\n<ul>\r\n\t<li>E[ 1 ] = 2020 % 16 = 4, X[ 1 ] = 4 \/ 4 = 1, Y[ 1 ] = 4 % 4 = 0.<br \/>\r\n\tHence, you&#39;ll build a bridge to connect island 1 and island 0.<\/li>\r\n\t<li>E[ 2 ] = (4 * 2 + 3) % 16 = 11, X[ 2 ] = 11 \/ 4 = 2, Y[ 2 ] = 11 % 4 = 3.<br \/>\r\n\tYou&#39;ll build a bridge to connect island 2 and island 3.<\/li>\r\n\t<li>E[ 3 ] = (11 * 2 + 3) % 16 = 9, X[ 3 ] = 9 \/ 4 = 2, Y[ 3 ] = 9 % 4 = 1.<br \/>\r\n\tYou&#39;ll build a bridge to connect island 2 and island 1.<\/li>\r\n\t<li>After the third bridge is built, now you can reach any island from any island, so the answer is M = 3.<\/li>\r\n<\/ul>\r\n\r\n<p>In another example, suppose N = 4, Seed = 2020, A = 3, and B = 4.<\/p>\r\n\r\n<ul>\r\n\t<li>E[ 1 ] = 2020 % 16 = 4, X[ 1 ] = 4 \/ 4 = 1, Y[ 1 ] = 4 % 4 = 0.<br \/>\r\n\tHence, you&#39;ll build a bridge to connect island 1 and island 0.<\/li>\r\n\t<li>E[ 2 ] = (4 * 3&nbsp;+ 4) % 16 = 0, X[ 2 ] = 0&nbsp;\/ 4&nbsp;= 0, Y[ 2 ] = 0 % 4 = 0.&nbsp;<br \/>\r\n\tBecause X[2] = Y[2], you won&#39;t build any bridges.<\/li>\r\n\t<li>E[ 3 ] = (0&nbsp;* 3&nbsp;+ 4) % 16 = 4, X[ 3 ] = 4 \/ 4 = 1, Y[ 3 ] = 4 % 4 = 0.<br \/>\r\n\tIsland 1 and Island 0 are already connected by a bridge, so you&#39;ll skip.<\/li>\r\n\t<li>In this example, you won&#39;t be able to build any other bridge than the first one. Hence, there is no M that meets your goal.<\/li>\r\n<\/ul>\r\n\r\n<p>Given N, Seed, A, and B, compute the correct value of M. If no such M exists, output 0.<\/p>\r\n","input":"<p>The first line will contain an integer, T, the number of test cases.<\/p>\r\n\r\n<p>Each test case will contain four integers separated by a whitespace: N, Seed, A, and B.<\/p>\r\n","output":"<p>Output the smallest M that meets the conditions stated in the problem statement. If no such M exists, output 0.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English","limit":"<ul>\r\n\t<li>1 &le; T &le; 10<\/li>\r\n\t<li>2 &le; N &le; 1,000<\/li>\r\n\t<li>1 &le; Seed, A, B &le; 1,000,000,000&nbsp;<\/li>\r\n<\/ul>\r\n"}]