시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
2 초 128 MB 35 10 10 38.462%

문제

해시(hash)란 주어진 데이터를 하나의 수로 표현하는 것을 말한다. 예를 들어서 "1024"와 같은 문자열을 1024 라는 수로 생각하는 방법은 해시의 한 예이다. 또, 영단어에서 'a'부터 'z'까지를 1에서 26까지의 수로 생각하여 그 수들을 합한 것도 해시의 한 예이다. 이런 해시를 적용하면 "ab"와 같은 영단어는 3이라는 수가 되고, "ace"와 같은 영단어는 9라는 수가 된다.

당신은 그래프에 대해서 해시 함수를 하나 설계하였다. 당신이 설계한 해시 값은 그래프의 1번 정점에서 2번 정점까지 가는 모든 경로들의 경로 값의 최소공배수로 정의한다. 경로 값은 경로 상에 있는 모든 간선의 가중치들의 최대공약수로 정의한다. 물론 경로에서는 같은 정점이 두 번 이상 나와서는 안 된다.

예를 들어 위와 같은 그래프를 보자. 위의 그래프에서 1번 정점에서 2번 정점까지 가는 경로는 1-4-2, 1-3-2의 두 개가 있다. 각 경로의 경로 값은 2(16, 6의 최대공약수), 3(3, 9의 최대공약수)이 되고, 따라서 해시 값은 6(2, 3의 최소공배수)이 된다.

하나의 그래프가 주어졌을 때, 그 그래프의 해시 값(위에서 설계한 해시 함수를 이용하여 계산한)을 구해내는 프로그램을 작성하시오.

입력

첫째 줄에 그래프의 정점의 개수 N(2≤N≤30)이 주어진다. 다음 N개의 줄에는 N개의 음 아닌 정수로 대칭적인 인접 행렬이 주어진다. 0인 경우는 간선이 없을 때를 나타내며, 그 외의 간선의 가중치는 1이상 5,000이하의 자연수이다.

출력

첫째 줄에 해시 값을 출력한다. 이 값은 10진수로 1000자리를 넘지 않는다.

예제 입력 1

4
0 0 3 16
0 0 9 6
3 9 0 0
16 6 0 0

예제 출력 1

6
[{"problem_id":"2328","problem_lang":"0","title":"\uadf8\ub798\ud504\uc758 \ud574\uc2dc","description":"<p>\ud574\uc2dc(hash)\ub780 \uc8fc\uc5b4\uc9c4 \ub370\uc774\ud130\ub97c \ud558\ub098\uc758 \uc218\ub85c \ud45c\ud604\ud558\ub294 \uac83\uc744 \ub9d0\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4\uc11c &quot;1024&quot;\uc640 \uac19\uc740 \ubb38\uc790\uc5f4\uc744 1024 \ub77c\ub294 \uc218\ub85c \uc0dd\uac01\ud558\ub294 \ubc29\ubc95\uc740 \ud574\uc2dc\uc758 \ud55c \uc608\uc774\ub2e4. \ub610, \uc601\ub2e8\uc5b4\uc5d0\uc11c 'a'\ubd80\ud130 'z'\uae4c\uc9c0\ub97c 1\uc5d0\uc11c 26\uae4c\uc9c0\uc758 \uc218\ub85c \uc0dd\uac01\ud558\uc5ec \uadf8 \uc218\ub4e4\uc744 \ud569\ud55c \uac83\ub3c4 \ud574\uc2dc\uc758 \ud55c \uc608\uc774\ub2e4. \uc774\ub7f0 \ud574\uc2dc\ub97c \uc801\uc6a9\ud558\uba74 &quot;ab&quot;\uc640 \uac19\uc740 \uc601\ub2e8\uc5b4\ub294 3\uc774\ub77c\ub294 \uc218\uac00 \ub418\uace0, &quot;ace&quot;\uc640 \uac19\uc740 \uc601\ub2e8\uc5b4\ub294 9\ub77c\ub294 \uc218\uac00 \ub41c\ub2e4.<\/p>\r\n<p>\ub2f9\uc2e0\uc740 \uadf8\ub798\ud504\uc5d0 \ub300\ud574\uc11c \ud574\uc2dc \ud568\uc218\ub97c \ud558\ub098 \uc124\uacc4\ud558\uc600\ub2e4. \ub2f9\uc2e0\uc774 \uc124\uacc4\ud55c \ud574\uc2dc \uac12\uc740 \uadf8\ub798\ud504\uc758 1\ubc88 \uc815\uc810\uc5d0\uc11c 2\ubc88 \uc815\uc810\uae4c\uc9c0 \uac00\ub294 \ubaa8\ub4e0 \uacbd\ub85c\ub4e4\uc758 \uacbd\ub85c \uac12\uc758 \ucd5c\uc18c\uacf5\ubc30\uc218\ub85c \uc815\uc758\ud55c\ub2e4. \uacbd\ub85c \uac12\uc740 \uacbd\ub85c \uc0c1\uc5d0 \uc788\ub294 \ubaa8\ub4e0 \uac04\uc120\uc758 \uac00\uc911\uce58\ub4e4\uc758 \ucd5c\ub300\uacf5\uc57d\uc218\ub85c \uc815\uc758\ud55c\ub2e4. \ubb3c\ub860 \uacbd\ub85c\uc5d0\uc11c\ub294 \uac19\uc740 \uc815\uc810\uc774 \ub450 \ubc88 \uc774\uc0c1 \ub098\uc640\uc11c\ub294 \uc548 \ub41c\ub2e4.<\/p>\r\n<p><img width=\"268\" height=\"239\" alt=\"\" src=\"\/JudgeOnline\/upload\/201011\/ha.PNG\" \/><\/p>\r\n<p>\uc608\ub97c \ub4e4\uc5b4 \uc704\uc640 \uac19\uc740 \uadf8\ub798\ud504\ub97c \ubcf4\uc790. \uc704\uc758 \uadf8\ub798\ud504\uc5d0\uc11c 1\ubc88 \uc815\uc810\uc5d0\uc11c 2\ubc88 \uc815\uc810\uae4c\uc9c0 \uac00\ub294 \uacbd\ub85c\ub294 1-4-2, 1-3-2\uc758 \ub450 \uac1c\uac00 \uc788\ub2e4. \uac01 \uacbd\ub85c\uc758 \uacbd\ub85c \uac12\uc740 2(16, 6\uc758 \ucd5c\ub300\uacf5\uc57d\uc218), 3(3, 9\uc758 \ucd5c\ub300\uacf5\uc57d\uc218)\uc774 \ub418\uace0, \ub530\ub77c\uc11c \ud574\uc2dc \uac12\uc740 6(2, 3\uc758 \ucd5c\uc18c\uacf5\ubc30\uc218)\uc774 \ub41c\ub2e4.<\/p>\r\n<p>\ud558\ub098\uc758 \uadf8\ub798\ud504\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uadf8 \uadf8\ub798\ud504\uc758 \ud574\uc2dc \uac12(\uc704\uc5d0\uc11c \uc124\uacc4\ud55c \ud574\uc2dc \ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uacc4\uc0b0\ud55c)\uc744 \uad6c\ud574\ub0b4\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \uadf8\ub798\ud504\uc758 \uc815\uc810\uc758 \uac1c\uc218 N(2&le;N&le;30)\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \ub2e4\uc74c N\uac1c\uc758 \uc904\uc5d0\ub294 N\uac1c\uc758 \uc74c \uc544\ub2cc \uc815\uc218\ub85c \ub300\uce6d\uc801\uc778 \uc778\uc811 \ud589\ub82c\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. 0\uc778 \uacbd\uc6b0\ub294 \uac04\uc120\uc774 \uc5c6\uc744 \ub54c\ub97c \ub098\ud0c0\ub0b4\uba70, \uadf8 \uc678\uc758 \uac04\uc120\uc758 \uac00\uc911\uce58\ub294 1\uc774\uc0c1 5,000\uc774\ud558\uc758 \uc790\uc5f0\uc218\uc774\ub2e4.<\/p>","output":"<p>\uccab\uc9f8 \uc904\uc5d0 \ud574\uc2dc \uac12\uc744 \ucd9c\ub825\ud55c\ub2e4. \uc774 \uac12\uc740 10\uc9c4\uc218\ub85c 1000\uc790\ub9ac\ub97c \ub118\uc9c0 \uc54a\ub294\ub2e4.<\/p>","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"2328","problem_lang":"1","title":"Cow Math","description":"<p>Taking their cue from the builders of the USA&#39;s Interstate Highway system, the cows have introduced the Interpasture Path numbering system. They have already numbered the N (2 &lt;= N &lt;= 30) pastures with the integers 1..N and now are numbering each path between two pastures with its own distinct Interpasture Path number in the range 1..5000 &nbsp;(e.g., I-9 and I-16).&nbsp;<\/p>\r\n\r\n<p>In an example Interpasture Path map, four pastures numbered 1, 2, 3, and 4 are connected by Interpasture Paths I-3, I-6, I-9, and I-16:&nbsp;<\/p>\r\n\r\n<pre>\r\n                  4--&lt; I-6&gt;--2\r\n\r\n                 \/         \/\r\n\r\n             &lt; I-16&gt;     &lt; I-9&gt;   \r\n\r\n               \/         \/\r\n\r\n              1--&lt; I-3&gt;--3  \r\n<\/pre>\r\n\r\n<p>Bessie likes to walk from pasture 1 to pasture 2 on the nifty new Interpasture system. During each walk, she never visits the same pasture twice, so possible walks on the sample map above are 1-4-2 and 1-3-2.&nbsp;<\/p>\r\n\r\n<p>Over the years, Bessie has developed an amazing mathematical skill that she likes to exercise. During each walk, she enjoys finding the greatest common factor (GCF) of the Interpasture Paths that she traverses. For instance, the walk designated 1-4-2 touches I-16 and I-6 which have the greatest common factor of 2 (since 2 properly divides into 16 and 6 but no larger integer does).&nbsp;<\/p>\r\n\r\n<p>As she walks the pastures day after day, she takes all the possible routes from pasture 1 to pasture 2 and remembers each of the GCFs. After she has taken every possible walk once, she computes the least common multiple (LCM) of all the GCFs. For this example, the two GCF values are 2 and 3 (GCF(6,16)=2 and GCF(3,9)=3), so the LCM is 6.&nbsp;<\/p>\r\n\r\n<p>For large networks of paths, Bessie might get tired of all the walking, but she really wants to know the LCM for every map. Calculate that number for her.&nbsp;<\/p>\r\n","input":"<ul>\r\n\t<li>Line 1: N&nbsp;<\/li>\r\n\t<li>Lines 2..N+1: These N lines represent the symmetric Interpasture Path connectivity matrix of the pastures. Line L shows the connectivity between pasture L-1 and the other pastures with its N space-separated integers. The first integer on each line is the Interpasture Path number that connects pasture L-1 and and pasture 1; the second integer is the IP number connecting pasture L-1 and pasture 2; etc. If pasture A connects to pasture B, then pasture B connects to Pasture A. When no Interpasture Path is available, the integer is 0.&nbsp;<\/li>\r\n<\/ul>\r\n","output":"<p>A single line with a single integer that is the LCM of the GCFs of all the possible walks from pasture 1 to pasture 2. It is guaranteed that the answer will contain 1000&nbsp;or fewer digits.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]