문제
위 그림은 3ドル^2 \times 3^2$ 서로소 스도쿠의 예시이다.
서로소 스도쿠는 스도쿠의 변형으로 간단한 숫자 퍼즐이다. 서로소 스도쿠는 $N^2 \times N^2$ 크기의 보드에서 진행되며 각 칸에는 정수 하나를 쓸 수 있다. 이 보드를 $N \times N$ 크기의 정사각 부분으로 나눈 $N^2$개의 영역들을 '블록'이라고 한다. 서로소 스도쿠의 목표는 다음 규칙을 만족하도록 빈칸에 수를 작성하는 것이다.
모든 수는 2ドル$ 이상 1ドル,000円,000円$ 이하의 정수이다.
한 행에서 서로 다른 두 위치에 적힌 두 수 $x_1, x_2$는 서로소이다. 즉 $\gcd(x_1, x_2) = 1$이다.
한 열에서 서로 다른 두 위치에 적힌 두 수 $x_1, x_2$는 서로소이다. 즉 $\gcd(x_1, x_2) = 1$이다.
한 블록에서 서로 다른 두 위치에 적힌 두 수 $x_1, x_2$는 서로소이다. 즉 $\gcd(x_1, x_2) = 1$이다.
일부 칸이 작성된 서로소 스도쿠를 입력받았을 때, 규칙에 맞게 모든 빈칸에 수를 채우는 프로그램을 작성하시오.
출력
$N^2$개의 줄에 걸쳐 모든 빈칸을 채운 서로소 스도쿠를 출력한다. 그중 $i$번째 줄에는 $N^2$개의 정수를 공백으로 구분하여 출력한다. 이 중 $j$번째 수 $b_{ij}$는 $i$번째 행 $j$번째 열에 있는 수를 의미한다. $(2 \le b_{ij} \le 1,000円,000円)$
서로소 스도쿠의 모든 칸을 채우는 방법이 여럿인 경우는 그중 아무거나 하나를 출력한다.
예제 출력 1
복사
4 77 5 13
15 13 8 11
49 9 13 10
13 2 7 3
[{"problem_id":"29759","problem_lang":"0","title":"\uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0","description":"<p style=\"text-align:center;max-width:1000px; margin: 0 auto\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/1be8e396-7b55-4c91-8bdf-ae3244ce2351\/-\/preview\/\" style=\"height: auto; width: 1000px;\" \/><\/p>\r\n\r\n<p style=\"text-align: center;\">\uc704 \uadf8\ub9bc\uc740 $3^2 \\times 3^2$ \uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0\uc758 \uc608\uc2dc\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0\ub294 \uc2a4\ub3c4\ucfe0\uc758 \ubcc0\ud615\uc73c\ub85c \uac04\ub2e8\ud55c \uc22b\uc790 \ud37c\uc990\uc774\ub2e4. \uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0\ub294 $N^2 \\times N^2$ \ud06c\uae30\uc758 \ubcf4\ub4dc\uc5d0\uc11c \uc9c4\ud589\ub418\uba70 \uac01 \uce78\uc5d0\ub294 \uc815\uc218 \ud558\ub098\ub97c \uc4f8 \uc218 \uc788\ub2e4. \uc774 \ubcf4\ub4dc\ub97c $N \\times N$ \ud06c\uae30\uc758 \uc815\uc0ac\uac01 \ubd80\ubd84\uc73c\ub85c \ub098\ub208 $N^2$\uac1c\uc758 \uc601\uc5ed\ub4e4\uc744 &#39;\ube14\ub85d&#39;\uc774\ub77c\uace0 \ud55c\ub2e4. \uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0\uc758 \ubaa9\ud45c\ub294 \ub2e4\uc74c \uaddc\uce59\uc744 \ub9cc\uc871\ud558\ub3c4\ub85d \ube48\uce78\uc5d0 \uc218\ub97c \uc791\uc131\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\ubaa8\ub4e0 \uc218\ub294 $2$ \uc774\uc0c1 $1\\,000\\,000$ \uc774\ud558\uc758 \uc815\uc218\uc774\ub2e4.<\/li>\r\n\t<li>\ud55c \ud589\uc5d0\uc11c \uc11c\ub85c \ub2e4\ub978 \ub450 \uc704\uce58\uc5d0 \uc801\ud78c \ub450 \uc218 $x_1, x_2$\ub294 \uc11c\ub85c\uc18c\uc774\ub2e4. \uc989 $\\gcd(x_1, x_2) = 1$\uc774\ub2e4.<\/li>\r\n\t<li>\ud55c \uc5f4\uc5d0\uc11c \uc11c\ub85c \ub2e4\ub978 \ub450 \uc704\uce58\uc5d0 \uc801\ud78c \ub450 \uc218 $x_1, x_2$\ub294 \uc11c\ub85c\uc18c\uc774\ub2e4. \uc989 $\\gcd(x_1, x_2) = 1$\uc774\ub2e4.<\/li>\r\n\t<li>\ud55c \ube14\ub85d\uc5d0\uc11c \uc11c\ub85c \ub2e4\ub978 \ub450 \uc704\uce58\uc5d0 \uc801\ud78c \ub450 \uc218 $x_1, x_2$\ub294 \uc11c\ub85c\uc18c\uc774\ub2e4. \uc989 $\\gcd(x_1, x_2) = 1$\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc77c\ubd80 \uce78\uc774 \uc791\uc131\ub41c \uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0\ub97c \uc785\ub825\ubc1b\uc558\uc744 \ub54c, \uaddc\uce59\uc5d0 \ub9de\uac8c \ubaa8\ub4e0 \ube48\uce78\uc5d0 \uc218\ub97c \ucc44\uc6b0\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab \ubc88\uc9f8 \uc904\uc5d0 $N$\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. $(2 \\le N \\le 10)$<\/p>\r\n\r\n<p>\ub450 \ubc88\uc9f8 \uc904\ubd80\ud130 $N^2$\uac1c\uc758 \uc904\uc5d0 \uac78\uccd0 \uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0 \ubcf4\ub4dc\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uadf8\uc911 $i$\ubc88\uc9f8 \uc904\uc5d0\ub294 $N^2$\uac1c\uc758 \uc815\uc218\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. \uc774 \uc911 $j$\ubc88\uc9f8 \uc218 $a_{ij}$\ub294 $i$\ubc88\uc9f8 \ud589 $j$\ubc88\uc9f8 \uc5f4\uc5d0 \uc788\ub294 \uc218\ub97c \uc758\ubbf8\ud55c\ub2e4. $a_{ij}$\uac00 $0$\uc778 \uacbd\uc6b0\ub294 $i$\ubc88\uc9f8 \ud589 $j$\ubc88\uc9f8 \uc5f4\uc774 \ube48\uce78\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4. $(0 \\le a_{ij} \\le 1\\,000\\,000;$ $a_{ij} \\neq 1)$<\/p>\r\n\r\n<p>\uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c0\ub294 \uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0\ub294 \ud56d\uc0c1 \uaddc\uce59\uc744 \ub9cc\uc871\ud558\uba70 \ube48\uce78\uc774 \ud56d\uc0c1 $1$\uac1c \uc774\uc0c1\uc774\ub2e4. \ubaa8\ub4e0 \ube48\uce78\uc744 \ucc44\uc6b8 \uc218 \uc5c6\ub294 \uacbd\uc6b0\uc758 \uc785\ub825\uc740 \uc8fc\uc5b4\uc9c0\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","output":"<p>$N^2$\uac1c\uc758 \uc904\uc5d0 \uac78\uccd0 \ubaa8\ub4e0 \ube48\uce78\uc744 \ucc44\uc6b4 \uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0\ub97c \ucd9c\ub825\ud55c\ub2e4. \uadf8\uc911 $i$\ubc88\uc9f8 \uc904\uc5d0\ub294 $N^2$\uac1c\uc758 \uc815\uc218\ub97c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud558\uc5ec \ucd9c\ub825\ud55c\ub2e4. \uc774 \uc911 $j$\ubc88\uc9f8 \uc218 $b_{ij}$\ub294 $i$\ubc88\uc9f8 \ud589 $j$\ubc88\uc9f8 \uc5f4\uc5d0 \uc788\ub294 \uc218\ub97c \uc758\ubbf8\ud55c\ub2e4. $(2 \\le b_{ij} \\le 1\\,000\\,000)$<\/p>\r\n\r\n<p>\uc11c\ub85c\uc18c \uc2a4\ub3c4\ucfe0\uc758 \ubaa8\ub4e0 \uce78\uc744 \ucc44\uc6b0\ub294 \ubc29\ubc95\uc774 \uc5ec\ub7ff\uc778 \uacbd\uc6b0\ub294 \uadf8\uc911 \uc544\ubb34\uac70\ub098 \ud558\ub098\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"29759","problem_lang":"1","title":"Coprime Sudoku","description":"<p style=\"text-align:center;max-width:1000px; margin: 0 auto\"><img alt=\"\" src=\"https:\/\/upload.acmicpc.net\/45c61358-accb-4a6d-a15a-999d9ed8b7e7\/-\/preview\/\" style=\"height: auto; width: 1000px;\" \/><\/p>\r\n\r\n<p style=\"text-align: center;\">The image above exemplifies a $3^2 \\times 3^2$ coprime sudoku.<\/p>\r\n\r\n<p>Coprime Sudoku is a simple variation of the traditional Sudoku puzzle. It&#39;s played on an $N^2 \\times N^2$ board, and each cell can contain one integer. This board is divided into $N^2$ areas of $N \\times N$ cells called &#39;blocks.&#39; The goal of Coprime Sudoku is to fill the empty cells following these rules:<\/p>\r\n\r\n<ul>\r\n\t<li>Every number is an integer between $2$&nbsp;and $1\\,000\\,000$ (inclusive).<\/li>\r\n\t<li>For two numbers $x_1, x_2$ written in different positions in the same row, $\\gcd(x_1, x_2) = 1$.<\/li>\r\n\t<li>For two numbers $x_1, x_2$\u200b written in different positions in the same column, $\\gcd(x_1, x_2) = 1$.<\/li>\r\n\t<li>For two numbers $x_1, x_2$ written in different positions in the same block, $\\gcd(x_1, x_2) = 1$.<\/li>\r\n<\/ul>\r\n\r\n<p>Given a partially filled Coprime Sudoku, write a program that fills all the empty cells following the rules.<\/p>\r\n","input":"<p>The first line contains the integer $N$.&nbsp;$(2 \\le N \\le 10)$<\/p>\r\n\r\n<p>From the second line to $N^2$ lines, the Coprime Sudoku board is given. The $i$-th line contains $N^2$ integers separated by spaces. The $j$-th number $a_{ij}$\u200b represents the number in the $i$-th row and $j$-th column. If $a_{ij}\u200b=0,$ it means the cell in the $i$-th row and $j$-th column is empty. $(0 \\le a_{ij} \\le 1\\,000\\,000;$ $a_{ij} \\neq 1)$<\/p>\r\n\r\n<p>The input will always follow the rules of Coprime Sudoku, and there will always be at least one empty cell. There are no inputs for which filling the empty cells is impossible.<\/p>\r\n","output":"<p>Output the Coprime Sudoku over $N^2$ lines, filling all the empty cells. The $i$-th line should contain $N^2$ integers separated by spaces. The $j$-th number $b_{ij}$\u200b represents the number in the $i$-th row and $j$-th column. $(2 \\le b_{ij} \\le 1\\,000\\,000)$<\/p>\r\n\r\n<p>If there are multiple solutions to the Coprime Sudoku, you can output any of them.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English"}]