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문제

호그볼트 수리마법과에 재학중인 서녕미온느는 올해 학교를 졸업한다. 상근 포터는 서녕미온느를 매우 심하게 괴롭혔고, 서녕미온느는 졸업을 하기 전에 상근 포터의 컴퓨터를 해킹하려고 한다.

상근 포터의 비밀번호는 1부터 N까지 자연수로 이루어진 순열이다. 상근 포터는 자연수 K에 집착한다. 예를 들어, 처음 보는 사람을 만났을 때, 그는 그 사람에게 자기 이름을 K번 말한다. 서녕미온느는 상근 포터가 비밀번호 치는 장면을 대학 재학동안 꾸준히 봤고, 그의 비밀번호의 순서는 K라는 사실을 알게되었다.

x ∈ {1, ..., N}의 모든 x를 상근 포터의 비밀번호에서 x가 있는 위치의 수로 K번 바꿨을 때, 다시 원래 x로 돌아오는 가장 작은 K를 순열의 순서라고 한다. 예를 들어, 순열이 2 3 1인 경우 1 → 3 → 2 → 1, 2 → 1 → 3 → 2, 3 → 2 → 1 → 3이기 때문에, 순열의 순서는 3이다.

N과 K가 주어졌을 때, 상근이의 비밀번호로 가능한 순열의 수를 구하는 프로그램을 작성하시오.

입력

첫째 줄에 N과 K가 주어진다. (1 ≤ N ≤ 100, 1 ≤ K ≤ 231 - 1)

출력

1부터 N까지 숫자로 이루어진 순열 중 순서가 K인 것의 개수를 231 - 1로 나눈 나머지를 출력한다. 

예제 입력 1

15 12

예제 출력 1

1789014075

힌트

[{"problem_id":"5041","problem_lang":"0","title":"\ud638\uadf8\ubcfc\ud2b8","description":"<p>\ud638\uadf8\ubcfc\ud2b8 \uc218\ub9ac\ub9c8\ubc95\uacfc\uc5d0 \uc7ac\ud559\uc911\uc778 \uc11c\ub155\ubbf8\uc628\ub290\ub294 \uc62c\ud574 \ud559\uad50\ub97c \uc878\uc5c5\ud55c\ub2e4. \uc0c1\uadfc \ud3ec\ud130\ub294 \uc11c\ub155\ubbf8\uc628\ub290\ub97c \ub9e4\uc6b0 \uc2ec\ud558\uac8c \uad34\ub86d\ud614\uace0, \uc11c\ub155\ubbf8\uc628\ub290\ub294 \uc878\uc5c5\uc744 \ud558\uae30 \uc804\uc5d0 \uc0c1\uadfc \ud3ec\ud130\uc758 \ucef4\ud4e8\ud130\ub97c \ud574\ud0b9\ud558\ub824\uace0 \ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc0c1\uadfc \ud3ec\ud130\uc758 \ube44\ubc00\ubc88\ud638\ub294 1\ubd80\ud130 N\uae4c\uc9c0 \uc790\uc5f0\uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc21c\uc5f4\uc774\ub2e4. \uc0c1\uadfc \ud3ec\ud130\ub294 \uc790\uc5f0\uc218 K\uc5d0 \uc9d1\ucc29\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \ucc98\uc74c \ubcf4\ub294 \uc0ac\ub78c\uc744 \ub9cc\ub0ac\uc744 \ub54c, \uadf8\ub294 \uadf8 \uc0ac\ub78c\uc5d0\uac8c \uc790\uae30 \uc774\ub984\uc744 K\ubc88 \ub9d0\ud55c\ub2e4. \uc11c\ub155\ubbf8\uc628\ub290\ub294 \uc0c1\uadfc \ud3ec\ud130\uac00 \ube44\ubc00\ubc88\ud638 \uce58\ub294 \uc7a5\uba74\uc744 \ub300\ud559 \uc7ac\ud559\ub3d9\uc548 \uafb8\uc900\ud788 \ubd24\uace0, \uadf8\uc758 \ube44\ubc00\ubc88\ud638\uc758 \uc21c\uc11c\ub294 K\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc54c\uac8c\ub418\uc5c8\ub2e4.<\/p>\r\n\r\n<p>x &isin; {1, ..., N}\uc758 \ubaa8\ub4e0 x\ub97c \uc0c1\uadfc \ud3ec\ud130\uc758 \ube44\ubc00\ubc88\ud638\uc5d0\uc11c x\uac00 \uc788\ub294 \uc704\uce58\uc758 \uc218\ub85c K\ubc88 \ubc14\uafe8\uc744 \ub54c, \ub2e4\uc2dc \uc6d0\ub798 x\ub85c \ub3cc\uc544\uc624\ub294 \uac00\uc7a5 \uc791\uc740 K\ub97c \uc21c\uc5f4\uc758 \uc21c\uc11c\ub77c\uace0 \ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc21c\uc5f4\uc774 2 3 1\uc778 \uacbd\uc6b0 1 &rarr; 3 &rarr; 2 &rarr; 1, 2 &rarr; 1 &rarr; 3 &rarr; 2, 3 &rarr; 2 &rarr; 1 &rarr; 3\uc774\uae30 \ub54c\ubb38\uc5d0, \uc21c\uc5f4\uc758 \uc21c\uc11c\ub294 3\uc774\ub2e4.<\/p>\r\n\r\n<p>N\uacfc K\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uc0c1\uadfc\uc774\uc758 \ube44\ubc00\ubc88\ud638\ub85c \uac00\ub2a5\ud55c \uc21c\uc5f4\uc758 \uc218\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 N\uacfc K\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &le; N &le; 100, 1 &le; K &le; 2<sup>31<\/sup> - 1)<\/p>\r\n","output":"<p>1\ubd80\ud130 N\uae4c\uc9c0 \uc22b\uc790\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc21c\uc5f4 \uc911 \uc21c\uc11c\uac00 K\uc778 \uac83\uc758 \uac1c\uc218\ub97c 2<sup>31<\/sup> - 1\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud55c\ub2e4.&nbsp;<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"5041","problem_lang":"1","title":"Code Permutations","description":"<p>You are soon to graduate from the mathemagician school of Hagworts, and you&#39;re quite content with yourself; all the hard work and elbowing has finally paid off. Being successful at most of you endeavors you would normally end up a herald of sound reasoning and good mathemagics. You, however, are different; not only have you spent your young years secretly hacking away at computers, writing code to do your assigned routine homework for you, but of late you have started planning how to cram all your mathemagical skills into a computer to completely eliminate the need for mathemagicians! To show the others just how great a visionary you are, you plan to make your graduation ceremony something they will never forget.<\/p>\r\n\r\n<p>To do this you need to break into the safe of your arch-nemesis, Hairy Peter. The safe is locked by a code mechanism: All natural numbers from 1 to N need to be typed in in the correct order, set by Hairy Peter. Fortunately you know that Hairy, being a good mathemagician, has a certain weakness; he has a rather unhealthy obsession with the number K. (For instance he always has to introduce himself K times whenever he meets new people, making him quite annoying to be around.) Thus you are certain that his code, when viewed as a permutation of the N first naturals, has order exactly K. (i.e. K is the smallest positive number such that if you K times replace x &isin; {1, ..., N} with the position of x in Hairy&#39;s code, you end up with the x you started with, for all x. Thus e.g. the order of the permutation corresponding to the code 2 3 1 is 3, as 1 &rarr; 3 &rarr; 2 &rarr; 1 and 2 &rarr; 1 &rarr; 3 &rarr; 2 and 3 &rarr; 2 &rarr; 1 &rarr; 3.) While this does not help you directly, it greatly reduces the number of code inputs you may have to try before you find the correct one. &quot;How many?&quot; is the question you are pondering right now. You must know the exact number, lest you risk preparing too little time for cracking the safe.<\/p>\r\n\r\n<p>Now you also have a certain mathemagical weakness arguably a bit worse than Hairy Peter&#39;s: Because of your dark scheme to program mathemagical computers, you insist there are no numbers greater than what can be represented by a signed 32-bit integer, namely the prime P = 2<sup>31<\/sup> - 1.Of course there must be nothing your computers cannot count. In fact you hate this upper limit P so intensely that you have decided to make a new mathemagics where P equals 0. Ha, take that! (Well, you are quite aware that you are really just counting modulo P, but it will have to suffice until you find better ways of punishing P.) In fact this turns out to be quite an advantage for you! For instance, if the number of code permutations you have to check turns out to be 2<sup>31<\/sup>, there will actually be just one permutation for you to check, as 2<sup>31<\/sup> mod P = 1. (Or at least you think so...) That&#39;s just magnificent!<\/p>\r\n","input":"<p>The input consists of two integers N (1 &le; N &le; 100) and K (1 &le; K &le; 2<sup>31<\/sup> - 1) respectively.<\/p>\r\n","output":"<p>The number of permutations of N elements of order K, modulo 2<sup>31<\/sup> - 1.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]