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문제

1926년 10월 9일의 신문엔 미국의 유명 극작가 벤 윌리암스의 짤막한 문제 하나가 실렸다. 전문은 다음과 같다.

다섯 명의 남자가 무인도에 갇혔다. 그들은 표류 첫 날, 하루 종일 힘을 합쳐 코코넛을 모았다.

그날 밤 첫 사람이 일어나 코코넛을 세어보니 하나를 빼면 정확히 5등분을 할 수 있었다.

그래서 지나가던 원숭이에게 코코넛 하나를 주고 나머지를 5등분하여 자기 몫의 한 무더기를 숨겨 두고 잠들었다.

그리고 바로 그 직후, 두 번째 사람이 일어나 코코넛을 세어보니 하나를 빼면 정확히 5등분을 할 수 있었다.

그래서 지나가던 원숭이에게 코코넛 하나를 주고 나머지를 5등분하여 자기 몫의 한 무더기를 숨겨 두고 잠들었다.

그 바로 다음 세 번째 사람도, 네 번째 사람도, 다섯 번째 사람도 같은 일을 했다.

이제 잠에서 깬 다섯 명이 남은 코코넛을 세어 보니 정확히 5등분을 할 수 있었다.

그래서 그들은 남은 코코넛을 5등분하여 한 묶음씩 가졌다.

이때, 그들이 처음 모은 코코넛은 모두 몇 개였을까?

문제의 답은 사실 무수히 많다. 하지만 그 중 가장 작은 수는 3121개이다.

하지만 이 문제는 우리가 풀 문제가 아니다.

우리는 코코넛 이야기를 반대로 생각해보기로 하자.

처음 모은 코코넛이 N개였고, 위의 규칙대로 K명의 사람들이 코코넛을 다들 나누어 가지는 데 성공했다면, 이때 K는 최대 몇이 될 수 있을까?

입력

입력은 여러 테스트 케이스로 이루어져 있다.

각 테스트 케이스의 첫 줄엔 문제의 N이 주어지며, N이 -1일 경우 입력의 종료를 의미한다.

출력

각 N에 따라 한 줄에, K가 존재할 경우 "N coconuts, max(K) people and 1 monkey" 를,

어떤 K도 코코넛을 규칙대로 나눌 수 없을 경우 "N coconuts, no solution"을 출력한다.

제한

  • 1 ≤ N ≤ 1,000,000

예제 입력 1

25
30
3121
-1

예제 출력 1

25 coconuts, 3 people and 1 monkey
30 coconuts, no solution
3121 coconuts, 5 people and 1 monkey
[{"problem_id":"6417","problem_lang":"0","title":"\ucf54\ucf54\ub11b \uadf8 \ub450 \ubc88\uc9f8 \uc774\uc57c\uae30","description":"<p>1926\ub144 10\uc6d4 9\uc77c\uc758 \uc2e0\ubb38\uc5d4 \ubbf8\uad6d\uc758 \uc720\uba85 \uadf9\uc791\uac00 \ubca4 \uc70c\ub9ac\uc554\uc2a4\uc758 \uc9e4\ub9c9\ud55c \ubb38\uc81c \ud558\ub098\uac00 \uc2e4\ub838\ub2e4. \uc804\ubb38\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p><strong>\ub2e4\uc12f \uba85\uc758 \ub0a8\uc790\uac00 \ubb34\uc778\ub3c4\uc5d0 \uac07\ud614\ub2e4. \uadf8\ub4e4\uc740 \ud45c\ub958 \uccab \ub0a0, \ud558\ub8e8 \uc885\uc77c \ud798\uc744 \ud569\uccd0 \ucf54\ucf54\ub11b\uc744 \ubaa8\uc558\ub2e4. <\/strong><\/p>\r\n\r\n<p><strong>\uadf8\ub0a0 \ubc24 \uccab \uc0ac\ub78c\uc774 \uc77c\uc5b4\ub098 \ucf54\ucf54\ub11b\uc744 \uc138\uc5b4\ubcf4\ub2c8 \ud558\ub098\ub97c \ube7c\uba74 \uc815\ud655\ud788 5\ub4f1\ubd84\uc744 \ud560 \uc218 \uc788\uc5c8\ub2e4.<\/strong><\/p>\r\n\r\n<p><strong>\uadf8\ub798\uc11c \uc9c0\ub098\uac00\ub358 \uc6d0\uc22d\uc774\uc5d0\uac8c \ucf54\ucf54\ub11b \ud558\ub098\ub97c \uc8fc\uace0 \ub098\uba38\uc9c0\ub97c 5\ub4f1\ubd84\ud558\uc5ec \uc790\uae30 \ubaab\uc758 \ud55c \ubb34\ub354\uae30\ub97c \uc228\uaca8 \ub450\uace0 \uc7a0\ub4e4\uc5c8\ub2e4. <\/strong><\/p>\r\n\r\n<p><strong>\uadf8\ub9ac\uace0 \ubc14\ub85c \uadf8 \uc9c1\ud6c4, \ub450 \ubc88\uc9f8 \uc0ac\ub78c\uc774 \uc77c\uc5b4\ub098 \ucf54\ucf54\ub11b\uc744 \uc138\uc5b4\ubcf4\ub2c8 \ud558\ub098\ub97c \ube7c\uba74 \uc815\ud655\ud788 5\ub4f1\ubd84\uc744 \ud560 \uc218 \uc788\uc5c8\ub2e4.<\/strong><\/p>\r\n\r\n<p><strong>\uadf8\ub798\uc11c \uc9c0\ub098\uac00\ub358 \uc6d0\uc22d\uc774\uc5d0\uac8c \ucf54\ucf54\ub11b \ud558\ub098\ub97c \uc8fc\uace0 \ub098\uba38\uc9c0\ub97c 5\ub4f1\ubd84\ud558\uc5ec \uc790\uae30 \ubaab\uc758 \ud55c \ubb34\ub354\uae30\ub97c \uc228\uaca8 \ub450\uace0 \uc7a0\ub4e4\uc5c8\ub2e4. <\/strong><\/p>\r\n\r\n<p><strong>\uadf8 \ubc14\ub85c \ub2e4\uc74c \uc138 \ubc88\uc9f8 \uc0ac\ub78c\ub3c4, \ub124 \ubc88\uc9f8 \uc0ac\ub78c\ub3c4, \ub2e4\uc12f \ubc88\uc9f8 \uc0ac\ub78c\ub3c4 \uac19\uc740 \uc77c\uc744 \ud588\ub2e4. <\/strong><\/p>\r\n\r\n<p><strong>\uc774\uc81c \uc7a0\uc5d0\uc11c \uae6c \ub2e4\uc12f \uba85\uc774 \ub0a8\uc740 \ucf54\ucf54\ub11b\uc744 \uc138\uc5b4 \ubcf4\ub2c8 \uc815\ud655\ud788 5\ub4f1\ubd84\uc744 \ud560 \uc218 \uc788\uc5c8\ub2e4. <\/strong><\/p>\r\n\r\n<p><strong>\uadf8\ub798\uc11c \uadf8\ub4e4\uc740 \ub0a8\uc740 \ucf54\ucf54\ub11b\uc744 5\ub4f1\ubd84\ud558\uc5ec \ud55c \ubb36\uc74c\uc529 \uac00\uc84c\ub2e4. <\/strong><\/p>\r\n\r\n<p><strong>\uc774\ub54c, \uadf8\ub4e4\uc774 \ucc98\uc74c \ubaa8\uc740 \ucf54\ucf54\ub11b\uc740 \ubaa8\ub450 \uba87 \uac1c\uc600\uc744\uae4c?<\/strong><\/p>\r\n\r\n<p>\ubb38\uc81c\uc758 \ub2f5\uc740 \uc0ac\uc2e4 \ubb34\uc218\ud788 \ub9ce\ub2e4. \ud558\uc9c0\ub9cc \uadf8 \uc911 \uac00\uc7a5 \uc791\uc740 \uc218\ub294 3121\uac1c\uc774\ub2e4.<\/p>\r\n\r\n<p>\ud558\uc9c0\ub9cc \uc774 \ubb38\uc81c\ub294 \uc6b0\ub9ac\uac00 \ud480 \ubb38\uc81c\uac00 \uc544\ub2c8\ub2e4.<\/p>\r\n\r\n<p>\uc6b0\ub9ac\ub294 \ucf54\ucf54\ub11b \uc774\uc57c\uae30\ub97c \ubc18\ub300\ub85c \uc0dd\uac01\ud574\ubcf4\uae30\ub85c \ud558\uc790.<\/p>\r\n\r\n<p>\ucc98\uc74c \ubaa8\uc740 \ucf54\ucf54\ub11b\uc774 N\uac1c\uc600\uace0, \uc704\uc758 \uaddc\uce59\ub300\ub85c K\uba85\uc758 \uc0ac\ub78c\ub4e4\uc774 \ucf54\ucf54\ub11b\uc744 \ub2e4\ub4e4 \ub098\ub204\uc5b4 \uac00\uc9c0\ub294 \ub370 \uc131\uacf5\ud588\ub2e4\uba74, \uc774\ub54c K\ub294 \ucd5c\ub300 \uba87\uc774 \ub420 \uc218 \uc788\uc744\uae4c?<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uccab \uc904\uc5d4 \ubb38\uc81c\uc758 N\uc774 \uc8fc\uc5b4\uc9c0\uba70, N\uc774 -1\uc77c \uacbd\uc6b0 \uc785\ub825\uc758 \uc885\ub8cc\ub97c \uc758\ubbf8\ud55c\ub2e4.<\/p>\r\n","output":"<p>\uac01 N\uc5d0 \ub530\ub77c \ud55c \uc904\uc5d0, K\uac00 \uc874\uc7ac\ud560 \uacbd\uc6b0 &quot;N coconuts, max(K) people and 1 monkey&quot; \ub97c,<\/p>\r\n\r\n<p>\uc5b4\ub5a4 K\ub3c4 \ucf54\ucf54\ub11b\uc744 \uaddc\uce59\ub300\ub85c \ub098\ub20c \uc218 \uc5c6\uc744 \uacbd\uc6b0 &quot;N coconuts, no solution&quot;\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"Korean","limit":"<ul>\r\n\t<li>1 &le; N &le; 1,000,000<\/li>\r\n<\/ul>\r\n"},{"problem_id":"6417","problem_lang":"1","title":"Coconuts, Revisited","description":"<p>The short story titled Coconuts, by Ben Ames Williams, appeared in the Saturday Evening Post on October 9, 1926. The story tells about five men and a monkey who were shipwrecked on an island. They spent the first night gathering coconuts. During the night, one man woke up and decided to take his share of the coconuts. He divided them into five piles. One coconut was left over so he gave it to the monkey, then hid his share and went back to sheep.<\/p>\r\n\r\n<p>Soon a second man woke up and did the same thing. After dividing the coconuts into five piles, one coconut was left over which he gave to the monkey. He then hid his share and went back to bed. The third, fourth, and fifth man followed exactly the same procedure. The next morning, after they all woke up, they divided the remaining coconuts into five equal shares. This time no coconuts were left over.<\/p>\r\n\r\n<p>An obvious question is &quot;how many coconuts did they originally gather?&quot; There are an infinite number of answers, but the lowest of these is 3,121. But that&#39;s not our problem here.<\/p>\r\n\r\n<p>Suppose we turn the problem around. If we know the number of coconuts that were gathered, what is the maximum number of persons (and one monkey) that could have been shipwrecked if the same procedure could occur?<\/p>\r\n","input":"<p>The input will consist of a sequence of integers, each representing the number of coconuts gathered by a group of persons (and a monkey) that were shipwrecked. The sequence will be followed by &quot;-1&quot;.<\/p>\r\n","output":"<p>For each number of coconuts, determine the largest number of persons who could have participated in the procedure described above. Display the results similar to the manner shown below, in the Expected Output. There may be no solution for some of the input cases; if so, state that observation.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English","limit":"<ul>\r\n\t<li>1 &le; N &le; 1,000,000<\/li>\r\n<\/ul>\r\n"}]

출처

ICPC > Regionals > North America > North Central North America Regional > NCNA 1997 3번