시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 128 MB 252 188 163 80.296%

문제

1960년, IBM의 직원 Donald Wall은 피보나치 수열을 m으로 나눈 나머지가 주기를 이룬다는 것을 증명했다.

예를 들어, 피보나치 수열의 처음 10개를 11로 나눈 예는 다음과 같다.

n 1 2 3 4 5 6 7 8 9 10
F(n) 1 1 2 3 5 8 13 21 34 55
F(n) mod 11 1 1 2 3 5 8 2 10 1 0

나머지를 이용해서 만든 수열은 주기가 나타날 수 있다. k(m)을 반복하는 부분 수열의 길이라고 했을 때, k(11) = 10임을 알 수 있다.

Wall은 아래와 같은 여러가지 성질도 증명했다.

  • m > 2인 경우에 k(m)은 짝수이다.
  • 임의의 짝수 정수 n > 2에 대해서, k(m) = n인 m이 항상 존재한다.
  • k(m) ≤ m2 - 1
  • k(2n) = 3×2(n-1)
  • k(5n) = 4×5n
  • k(2×5n) = 6n
  • n > 2라면, k(10n) = 15×10(n-1)

m이 주어졌을 때, k(m)을 구하는 프로그램을 작성하시오.

입력

첫째 줄에 테스트 케이스의 개수 P (1 ≤ P ≤ 1000)가 주어진다. 각 테스트 케이스는 N과 M으로 이루어져 있다. N은 테스트 케이스의 번호이고, M은 문제에서 설명한 m이다. (2 ≤ M ≤ 1,000,000)

출력

각 테스트 케이스마다 테스트 케이스 번호를 출력하고 k(M)을 출력한다.

예제 입력 1

5
1 4
2 5
3 11
4 123456
5 987654

예제 출력 1

1 6
2 20
3 10
4 15456
5 332808

힌트

[{"problem_id":"9471","problem_lang":"0","title":"\ud53c\uc0ac\ub178 \uc8fc\uae30","description":"<p>1960\ub144, IBM\uc758 \uc9c1\uc6d0 Donald Wall\uc740 \ud53c\ubcf4\ub098\uce58 \uc218\uc5f4\uc744 m\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\uac00 \uc8fc\uae30\ub97c \uc774\ub8ec\ub2e4\ub294 \uac83\uc744 \uc99d\uba85\ud588\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, \ud53c\ubcf4\ub098\uce58 \uc218\uc5f4\uc758 \ucc98\uc74c 10\uac1c\ub97c 11\ub85c \ub098\ub208 \uc608\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<table class=\"table table-bordered\" style=\"width:60%\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th>n<\/th>\r\n\t\t\t<th>1<\/th>\r\n\t\t\t<th>2<\/th>\r\n\t\t\t<th>3<\/th>\r\n\t\t\t<th>4<\/th>\r\n\t\t\t<th>5<\/th>\r\n\t\t\t<th>6<\/th>\r\n\t\t\t<th>7<\/th>\r\n\t\t\t<th>8<\/th>\r\n\t\t\t<th>9<\/th>\r\n\t\t\t<th>10<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<th>F(n)<\/th>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>2<\/td>\r\n\t\t\t<td>3<\/td>\r\n\t\t\t<td>5<\/td>\r\n\t\t\t<td>8<\/td>\r\n\t\t\t<td>13<\/td>\r\n\t\t\t<td>21<\/td>\r\n\t\t\t<td>34<\/td>\r\n\t\t\t<td>55<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<th>F(n) mod 11<\/th>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>2<\/td>\r\n\t\t\t<td>3<\/td>\r\n\t\t\t<td>5<\/td>\r\n\t\t\t<td>8<\/td>\r\n\t\t\t<td>2<\/td>\r\n\t\t\t<td>10<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>0<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>\ub098\uba38\uc9c0\ub97c \uc774\uc6a9\ud574\uc11c \ub9cc\ub4e0 \uc218\uc5f4\uc740 \uc8fc\uae30\uac00 \ub098\ud0c0\ub0a0 \uc218 \uc788\ub2e4. k(m)\uc744 \ubc18\ubcf5\ud558\ub294 \ubd80\ubd84 \uc218\uc5f4\uc758 \uae38\uc774\ub77c\uace0 \ud588\uc744 \ub54c, k(11) = 10\uc784\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>Wall\uc740 \uc544\ub798\uc640 \uac19\uc740 \uc5ec\ub7ec\uac00\uc9c0 \uc131\uc9c8\ub3c4 \uc99d\uba85\ud588\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>m &gt; 2\uc778 \uacbd\uc6b0\uc5d0 k(m)\uc740 \uc9dd\uc218\uc774\ub2e4.<\/li>\r\n\t<li>\uc784\uc758\uc758 \uc9dd\uc218 \uc815\uc218 n &gt; 2\uc5d0 \ub300\ud574\uc11c, k(m) = n\uc778 m\uc774 \ud56d\uc0c1 \uc874\uc7ac\ud55c\ub2e4.<\/li>\r\n\t<li>k(m) &le; m<sup>2<\/sup> - 1<\/li>\r\n\t<li>k(2<sup>n<\/sup>) = 3&times;2<sup>(n-1)<\/sup><\/li>\r\n\t<li>k(5<sup>n<\/sup>) = 4&times;5<sup>n<\/sup><\/li>\r\n\t<li>k(2&times;5<sup>n<\/sup>) = 6n<\/li>\r\n\t<li>n &gt; 2\ub77c\uba74, k(10<sup>n<\/sup>) = 15&times;10<sup>(n-1)<\/sup><\/li>\r\n<\/ul>\r\n\r\n<p>m\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, k(m)\uc744 \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uac1c\uc218 P (1 &le; P &le; 1000)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 N\uacfc M\uc73c\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. N\uc740 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \ubc88\ud638\uc774\uace0, M\uc740 \ubb38\uc81c\uc5d0\uc11c \uc124\uba85\ud55c m\uc774\ub2e4. (2 &le; M &le; 1,000,000)<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4 \ubc88\ud638\ub97c \ucd9c\ub825\ud558\uace0 k(M)\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"9471","problem_lang":"1","title":"Pisano Periods","description":"<p>IN 1960, Donald Wall of IBM, in White Plains, NY, proved that the series obtained by taking each element of the Fibonacci series modulo m was periodic.<\/p>\r\n\r\n<p>For example, the first ten element of the Fibonacci sequence, as well as their remainders modulo 11, are:<\/p>\r\n\r\n<table class=\"table table-bordered\" style=\"width:60%\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th>n<\/th>\r\n\t\t\t<th>1<\/th>\r\n\t\t\t<th>2<\/th>\r\n\t\t\t<th>3<\/th>\r\n\t\t\t<th>4<\/th>\r\n\t\t\t<th>5<\/th>\r\n\t\t\t<th>6<\/th>\r\n\t\t\t<th>7<\/th>\r\n\t\t\t<th>8<\/th>\r\n\t\t\t<th>9<\/th>\r\n\t\t\t<th>10<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<th>F(n)<\/th>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>2<\/td>\r\n\t\t\t<td>3<\/td>\r\n\t\t\t<td>5<\/td>\r\n\t\t\t<td>8<\/td>\r\n\t\t\t<td>13<\/td>\r\n\t\t\t<td>21<\/td>\r\n\t\t\t<td>34<\/td>\r\n\t\t\t<td>55<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<th>F(n) mod 11<\/th>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>2<\/td>\r\n\t\t\t<td>3<\/td>\r\n\t\t\t<td>5<\/td>\r\n\t\t\t<td>8<\/td>\r\n\t\t\t<td>2<\/td>\r\n\t\t\t<td>10<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>0<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>The sequence made up of the remainders then repeats. Let k(m) be the length of the repeating subsequence; in this example, we see k(11) = 10.<\/p>\r\n\r\n<p>Wall proved several other properties, some of which you may find interesting:<\/p>\r\n\r\n<ul>\r\n\t<li>If m &gt; 2, k(m) is even.<\/li>\r\n\t<li>For any even integer&nbsp;n &gt; 2, there exists m such that&nbsp;k(m) = n.<\/li>\r\n\t<li>k(m) &le; m<sup>2<\/sup>&nbsp;- 1<\/li>\r\n\t<li>k(2<sup>n<\/sup>) = 3&times;2<sup>(n-1)<\/sup><\/li>\r\n\t<li>k(5<sup>n<\/sup>) = 4&times;5<sup>n<\/sup><\/li>\r\n\t<li>k(2&times;5<sup>n<\/sup>) = 6n<\/li>\r\n\t<li>If n &gt; 2, k(10<sup>n<\/sup>) = 15&times;10<sup>(n-1)<\/sup><\/li>\r\n<\/ul>\r\n\r\n<p>For this problem, you must write a program that calculates the length of the repeating subsequence, k(m), for different modulo values m.<\/p>\r\n","input":"<p>The first line of input contains a single integer P, (1 &le; P &le; 1000), which is the number of data sets that follow. Each data set is a single line that consists of two space separated integer values N and M. N is the data set number. M is the modulo value (2 &le; M &le; 1,000,000)<\/p>\r\n","output":"<p>For each data set there is one line of output. It contains tha data set number (N) followed by a single space, followed by the length of the repeating subsequence for M, k(M).<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]