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문제

고고학자들은 고대 문명이 남긴 이상한 기계를 발견하였다. 이 기계는 두 정수 x와 y를 출력하는 두 부분이 있다.

이 기계를 조사해보고, 고고학자들은 이 기계가 과거의 어느 시점부터 시작하여 시점 t에 대한 정보를 출력 하는 특별한 시계라는 결론을 내렸다. 시점 T에서 첫 번째 출력 부분에서는 정수 x = ((t + ⌊t/B⌋ ) mod A)를 출력하고, 두번째 출력 부분에서는 정수 y = (t mod B)를 출력한다. (⌊x⌋는 x 이하이면서 가장 큰 정수를 나타낸다.)

분석을 해 보니 이 기계가 항상 동작하는 것은 아니며, n개의 연속된 구간 [li, ri]에서만 동작하는 것을 알게 되었다. 향후 연구를 위해서, 고고학자들은 당신에게 이 기계가 출력하는 순서쌍 (x, y) 중 서로 다른 것이 모두 몇 개인지를 알아내는 프로그램을 작성하도록 부탁했다.

두 순서쌍 (x1, y1)와 (x2, y2)는 만약 x1 ≠ x2이거나 y1 ≠ y2이라면 서로 다르다.

입력

첫 줄에는 세 정수 n, A, B가 주어진다. (1 ≤ n ≤ 106; 1 ≤ A, B ≤ 1018).

다음 n줄의 각각에는 두 정수 li와 ri가 주어지는데, 이 기계가 동작하는 구간 [li, ri]의 시작 시점과 종료 시점을 나타낸다. (0 ≤ li ≤ ri ≤ 1018 , ri < li+1).

출력

이 기계가 동작하는 동안 출력하는 서로 다른 순서쌍 (x, y)의 수를 출력한다.

예제 입력 1

3 3 3
4 4
7 9
17 18

예제 출력 1

4

예제 입력 2

3 5 10
1 20
50 68
89 98

예제 출력 2

31

예제 입력 3

2 16 13
2 5
18 18

예제 출력 3

5

힌트

첫 번째 테스트에서, 이 기계는 시점 4에서 (2, 1)을, 시점 7에서 (0, 1)을, 시점 8에서 (1, 2)를, 시점 9 에서 (0, 0)를, 시점 17에서 (1, 2)를, 시점 18에서 (0, 0)를 출력한다. 따라서 서로 다른 네 개의 순서쌍 (0, 0),(0, 1),(1, 2),(2, 1)을 출력한다.

[{"problem_id":"17634","problem_lang":"0","title":"\uc774\uc0c1\ud55c \uae30\uacc4","description":"<p>\uace0\uace0\ud559\uc790\ub4e4\uc740 \uace0\ub300 \ubb38\uba85\uc774 \ub0a8\uae34 \uc774\uc0c1\ud55c \uae30\uacc4\ub97c \ubc1c\uacac\ud558\uc600\ub2e4. \uc774 \uae30\uacc4\ub294 \ub450 \uc815\uc218 x\uc640 y\ub97c \ucd9c\ub825\ud558\ub294 \ub450 \ubd80\ubd84\uc774 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uc774 \uae30\uacc4\ub97c \uc870\uc0ac\ud574\ubcf4\uace0, \uace0\uace0\ud559\uc790\ub4e4\uc740 \uc774 \uae30\uacc4\uac00 \uacfc\uac70\uc758 \uc5b4\ub290 \uc2dc\uc810\ubd80\ud130 \uc2dc\uc791\ud558\uc5ec \uc2dc\uc810 t\uc5d0 \ub300\ud55c \uc815\ubcf4\ub97c \ucd9c\ub825 \ud558\ub294 \ud2b9\ubcc4\ud55c \uc2dc\uacc4\ub77c\ub294 \uacb0\ub860\uc744 \ub0b4\ub838\ub2e4. \uc2dc\uc810 T\uc5d0\uc11c \uccab \ubc88\uc9f8 \ucd9c\ub825 \ubd80\ubd84\uc5d0\uc11c\ub294 \uc815\uc218 x = ((t + &lfloor;t\/B&rfloor; ) mod A)\ub97c \ucd9c\ub825\ud558\uace0, \ub450\ubc88\uc9f8 \ucd9c\ub825 \ubd80\ubd84\uc5d0\uc11c\ub294 \uc815\uc218 y = (t mod B)\ub97c \ucd9c\ub825\ud55c\ub2e4. (&lfloor;x&rfloor;\ub294 x \uc774\ud558\uc774\uba74\uc11c \uac00\uc7a5 \ud070 \uc815\uc218\ub97c \ub098\ud0c0\ub0b8\ub2e4.)<\/p>\r\n\r\n<p>\ubd84\uc11d\uc744 \ud574 \ubcf4\ub2c8 \uc774 \uae30\uacc4\uac00 \ud56d\uc0c1 \ub3d9\uc791\ud558\ub294 \uac83\uc740 \uc544\ub2c8\uba70, n\uac1c\uc758 \uc5f0\uc18d\ub41c \uad6c\uac04 [l<sub>i<\/sub>, r<sub>i<\/sub>]\uc5d0\uc11c\ub9cc \ub3d9\uc791\ud558\ub294 \uac83\uc744 \uc54c\uac8c \ub418\uc5c8\ub2e4. \ud5a5\ud6c4 \uc5f0\uad6c\ub97c \uc704\ud574\uc11c, \uace0\uace0\ud559\uc790\ub4e4\uc740 \ub2f9\uc2e0\uc5d0\uac8c \uc774 \uae30\uacc4\uac00 \ucd9c\ub825\ud558\ub294 \uc21c\uc11c\uc30d (x, y) \uc911 \uc11c\ub85c \ub2e4\ub978 \uac83\uc774 \ubaa8\ub450 \uba87 \uac1c\uc778\uc9c0\ub97c \uc54c\uc544\ub0b4\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\ub3c4\ub85d \ubd80\ud0c1\ud588\ub2e4.<\/p>\r\n\r\n<p>\ub450 \uc21c\uc11c\uc30d (x<sub>1<\/sub>, y<sub>1<\/sub>)\uc640 (x<sub>2<\/sub>, y<sub>2<\/sub>)\ub294 \ub9cc\uc57d x<sub>1<\/sub> &ne;&nbsp;x<sub>2<\/sub>\uc774\uac70\ub098 y<sub>1<\/sub> &ne;&nbsp;y<sub>2<\/sub>\uc774\ub77c\uba74 \uc11c\ub85c \ub2e4\ub974\ub2e4.<\/p>\r\n","input":"<p>\uccab \uc904\uc5d0\ub294 \uc138 \uc815\uc218 n, A, B\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &le; n &le; 10<sup>6<\/sup>; 1 &le; A, B &le; 10<sup>18<\/sup>).<\/p>\r\n\r\n<p>\ub2e4\uc74c n\uc904\uc758 \uac01\uac01\uc5d0\ub294 \ub450 \uc815\uc218 l<sub>i<\/sub>\uc640 r<sub>i<\/sub>\uac00 \uc8fc\uc5b4\uc9c0\ub294\ub370, \uc774 \uae30\uacc4\uac00 \ub3d9\uc791\ud558\ub294 \uad6c\uac04 [l<sub>i<\/sub>, r<sub>i<\/sub>]\uc758 \uc2dc\uc791 \uc2dc\uc810\uacfc \uc885\ub8cc \uc2dc\uc810\uc744 \ub098\ud0c0\ub0b8\ub2e4. (0 &le; l<sub>i<\/sub> &le; r<sub>i<\/sub> &le; 10<sup>18<\/sup> , r<sub>i<\/sub> &lt; l<sub>i+1<\/sub>).<\/p>\r\n","output":"<p>\uc774 \uae30\uacc4\uac00 \ub3d9\uc791\ud558\ub294 \ub3d9\uc548 \ucd9c\ub825\ud558\ub294 \uc11c\ub85c \ub2e4\ub978 \uc21c\uc11c\uc30d (x, y)\uc758 \uc218\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"<p>\uccab \ubc88\uc9f8 \ud14c\uc2a4\ud2b8\uc5d0\uc11c, \uc774 \uae30\uacc4\ub294 \uc2dc\uc810 4\uc5d0\uc11c (2, 1)\uc744, \uc2dc\uc810 7\uc5d0\uc11c (0, 1)\uc744, \uc2dc\uc810 8\uc5d0\uc11c (1, 2)\ub97c, \uc2dc\uc810 9 \uc5d0\uc11c (0, 0)\ub97c, \uc2dc\uc810 17\uc5d0\uc11c (1, 2)\ub97c, \uc2dc\uc810 18\uc5d0\uc11c (0, 0)\ub97c \ucd9c\ub825\ud55c\ub2e4. \ub530\ub77c\uc11c \uc11c\ub85c \ub2e4\ub978 \ub124 \uac1c\uc758 \uc21c\uc11c\uc30d (0, 0),(0, 1),(1, 2),(2, 1)\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","original":"1","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"17634","problem_lang":"1","title":"Strange Device","description":"<p>Archaeologists have found a strange device that was probably created by some ancient civilization. The device has a screen that displays two integers: x and y.<\/p>\r\n\r\n<p>After exploring the device the scientists have made a conclusion that the device is kind of a clock. It measures time t passed from some moment in the past, but shows it in some weird way, probably used by the creators of the device. If the time passed is an integer t, the two integers displayed are: x = ((t + &lfloor;t\/B&rfloor; ) mod A), and y = (t mod B). Here &lfloor;x&rfloor; is the floor function &mdash; the greatest integer less or equal to x.<\/p>\r\n\r\n<p>The archaeologists have studied the device and found out that its screen wasn&rsquo;t turned on all the time. Actually it was only working during n continuous periods of time, the i-th of them was from the moment l<sub>i<\/sub> to the moment r<sub>i<\/sub>, inclusive. Now the scientists would like to calculate how many distinct pairs (x, y) were shown by the device when its screen was on.<\/p>\r\n\r\n<p>Two pairs (x<sub>1<\/sub>, y<sub>1<\/sub>) and (x<sub>2<\/sub>, y<sub>2<\/sub>) are distinct if x<sub>1<\/sub> &ne; x<sub>2<\/sub> or y<sub>1<\/sub> &ne; y<sub>2<\/sub>.<\/p>\r\n","input":"<p>The first line contains three integers n, A, and B (1 &le; n &le; 10<sup>6<\/sup>; 1 &le; A, B &le; 10<sup>18<\/sup>).<\/p>\r\n\r\n<p>Each of the following n lines contains two integers l<sub>i<\/sub> and r<sub>i<\/sub>, the beginning and the end of the i-th segment [l<sub>i<\/sub>, r<sub>i<\/sub>] when the device screen was turned on (0 &le; l<sub>i<\/sub> &le; r<sub>i<\/sub> &le; 10<sup>18<\/sup>; r<sub>i<\/sub> &lt; l<sub>i+1<\/sub>).<\/p>\r\n","output":"<p>Output the number of distinct pairs (x, y) that were shown on the device screen when it was turned on.<\/p>\r\n","hint":"<p>In the first test, the device screen shows the following integers.<\/p>\r\n\r\n<table class=\"table table-bordered\" style=\"width:20%;\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th style=\"text-align: center;\">t<\/th>\r\n\t\t\t<th style=\"text-align: center;\">(x, y)<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align: center;\">4<\/td>\r\n\t\t\t<td style=\"text-align: center;\">(2, 1)<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align: center;\">7<\/td>\r\n\t\t\t<td style=\"text-align: center;\">(0, 1)<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align: center;\">8<\/td>\r\n\t\t\t<td style=\"text-align: center;\">(1, 2)<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align: center;\">9<\/td>\r\n\t\t\t<td style=\"text-align: center;\">(0, 0)<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align: center;\">17<\/td>\r\n\t\t\t<td style=\"text-align: center;\">(1, 2)<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td style=\"text-align: center;\">18<\/td>\r\n\t\t\t<td style=\"text-align: center;\">(0, 0)<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>So there are four distinct pairs (0, 0),(0, 1),(1, 2),(2, 1).<\/p>\r\n","original":"0","problem_lang_code":"\uc601\uc5b4"}]

출처

Olympiad > Asia-Pacific Informatics Olympiad > APIO 2019 A번