문제
Albert는 "단조 증가 수" (monotone increasing number)를 이용한 놀이를 즐겨한다. 음이 아닌 임의의 정수 $x$에 대해 $x$의 각 자리 숫자들을 좌측 (가장 큰 자릿수) 부터 우측 (가장 작은 1의 자릿수)까지 순서대로 읽었을 때 한 번도 감소하지 않으면 $x$를 단조 증가 수라 부른다. 예를 들어 0, 5, 111, 234, 2246 등은 단조 증가 수이며 121, 465 등은 단조 증가 수가 아니다.
Albert가 평소 즐겨하는 놀이는 임의의 정수 $x$가 주어졌을 때, $x$를 초과하지 않으면서 가장 큰 단조 증가 수를 찾는 놀이이다 -- 편의상 그러한 정수를 $S(x)$라 하자. 이 정의에 따르면 $x$가 단조 증가 수라면 당연히 $S(x) = x$이며 $S(5) = 5$와 $S(111) = 111$ 등이 이러한 경우에 해당한다. 반면, $x$가 단조 증가 수가 아닌 경우는 당연히 $S(x) \lt x$이며, $S(121) = 119$ 와 $S(465) = 459$ 등이 이러한 경우에 해당한다.
Albert의 놀이를 지켜보던 그의 누나는 조금 더 재미있는 놀이를 제안했다. 0 이상인 두 정수 $N \le M$이 주어졌을 때, $F(N, M) = \sum_{x=N}^{M} S(x)$ 를 구하는 놀이이다. 즉, $N$이상 $M$이하인 모든 정수에 $x$ 대해 $S(x)$를 구하여 더한 값이 $F(N, M)$이 된다. Albert와 그의 누나를 도와 이 값을 구해보자.
출력
각 줄에 각 테스트 케이스의 정답인 $F(N, M)$을 출력한다. 단, 이 값이 매우 클 수 있으므로 10ドル^9 + 7$로 나눈 나머지를 출력한다.
제한
1ドル \le T \le 1000$
0ドル \le N \le M \le 10^{18}$
예제 출력 1
복사
119
459
406
740694249
예제 1-2: 본문에서 다루었다.
예제 3: 42ドル \le x \le 50$인 $S(x)$의 값은 순서대로 39,ドル 39, 44, 45, 46, 47, 48, 49, 49$ 이므로 총합인 406이 정답이다.
예제 4: 답이 매우 커질 수 있으므로 10ドル^9 + 7$로 나눈 나머지를 출력함에 유의하자.
[{"problem_id":"27928","problem_lang":"0","title":"\ub2e8\uc870 \uc99d\uac00 \uc218","description":"<p>Albert\ub294 &quot;\ub2e8\uc870 \uc99d\uac00 \uc218&quot; (monotone increasing number)\ub97c \uc774\uc6a9\ud55c \ub180\uc774\ub97c \uc990\uaca8\ud55c\ub2e4. \uc74c\uc774 \uc544\ub2cc \uc784\uc758\uc758 \uc815\uc218 $x$\uc5d0 \ub300\ud574 $x$\uc758 \uac01 \uc790\ub9ac \uc22b\uc790\ub4e4\uc744 \uc88c\uce21 (\uac00\uc7a5 \ud070 \uc790\ub9bf\uc218) \ubd80\ud130 \uc6b0\uce21 (\uac00\uc7a5 \uc791\uc740 1\uc758 \uc790\ub9bf\uc218)\uae4c\uc9c0 \uc21c\uc11c\ub300\ub85c \uc77d\uc5c8\uc744 \ub54c \ud55c \ubc88\ub3c4 \uac10\uc18c\ud558\uc9c0 \uc54a\uc73c\uba74 $x$\ub97c \ub2e8\uc870 \uc99d\uac00 \uc218\ub77c \ubd80\ub978\ub2e4. \uc608\ub97c \ub4e4\uc5b4 0, 5, 111, 234, 2246 \ub4f1\uc740 \ub2e8\uc870 \uc99d\uac00 \uc218\uc774\uba70 121, 465 \ub4f1\uc740 \ub2e8\uc870 \uc99d\uac00 \uc218\uac00 \uc544\ub2c8\ub2e4.<\/p>\r\n\r\n<p>Albert\uac00 \ud3c9\uc18c \uc990\uaca8\ud558\ub294 \ub180\uc774\ub294 \uc784\uc758\uc758 \uc815\uc218 $x$\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, $x$\ub97c \ucd08\uacfc\ud558\uc9c0 \uc54a\uc73c\uba74\uc11c \uac00\uc7a5 \ud070 \ub2e8\uc870 \uc99d\uac00 \uc218\ub97c \ucc3e\ub294 \ub180\uc774\uc774\ub2e4 -- \ud3b8\uc758\uc0c1 \uadf8\ub7ec\ud55c \uc815\uc218\ub97c $S(x)$\ub77c \ud558\uc790. \uc774 \uc815\uc758\uc5d0 \ub530\ub974\uba74 $x$\uac00 \ub2e8\uc870 \uc99d\uac00 \uc218\ub77c\uba74 \ub2f9\uc5f0\ud788 $S(x) = x$\uc774\uba70 $S(5) = 5$\uc640 $S(111) = 111$ \ub4f1\uc774 \uc774\ub7ec\ud55c \uacbd\uc6b0\uc5d0 \ud574\ub2f9\ud55c\ub2e4. \ubc18\uba74, $x$\uac00 \ub2e8\uc870 \uc99d\uac00 \uc218\uac00 \uc544\ub2cc \uacbd\uc6b0\ub294 \ub2f9\uc5f0\ud788 $S(x) \\lt x$\uc774\uba70, $S(121) = 119$ \uc640 $S(465) = 459$ \ub4f1\uc774 \uc774\ub7ec\ud55c \uacbd\uc6b0\uc5d0 \ud574\ub2f9\ud55c\ub2e4.<\/p>\r\n\r\n<p>Albert\uc758 \ub180\uc774\ub97c \uc9c0\ucf1c\ubcf4\ub358 \uadf8\uc758 \ub204\ub098\ub294 \uc870\uae08 \ub354 \uc7ac\ubbf8\uc788\ub294 \ub180\uc774\ub97c \uc81c\uc548\ud588\ub2e4. 0 \uc774\uc0c1\uc778 \ub450 \uc815\uc218 $N \\le M$\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, $F(N, M) = \\sum_{x=N}^{M} S(x)$ \ub97c \uad6c\ud558\ub294 \ub180\uc774\uc774\ub2e4. \uc989, $N$\uc774\uc0c1 $M$\uc774\ud558\uc778 \ubaa8\ub4e0 \uc815\uc218\uc5d0 $x$ \ub300\ud574 $S(x)$\ub97c \uad6c\ud558\uc5ec \ub354\ud55c \uac12\uc774 $F(N, M)$\uc774 \ub41c\ub2e4. Albert\uc640 \uadf8\uc758 \ub204\ub098\ub97c \ub3c4\uc640 \uc774 \uac12\uc744 \uad6c\ud574\ubcf4\uc790.<\/p>\r\n","input":"<p>\uc785\ub825 \uccab \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc218 $T$ \uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\ub2e4\uc74c $T$\uc904\uc5d0 \uac78\uccd0 \uac01 \uc904\uc5d0 $N, M$\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \uc904\uc5d0 \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc815\ub2f5\uc778 $F(N, M)$\uc744 \ucd9c\ub825\ud55c\ub2e4. \ub2e8, \uc774 \uac12\uc774 \ub9e4\uc6b0 \ud074 \uc218 \uc788\uc73c\ubbc0\ub85c $10^9 + 7$\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean","limit":"<ul>\r\n\t<li>$1 \\le T \\le 1000$<\/li>\r\n\t<li>$0 \\le N \\le M \\le 10^{18}$<\/li>\r\n<\/ul>\r\n","sample_explain_1":"<p>\uc608\uc81c 1-2: \ubcf8\ubb38\uc5d0\uc11c \ub2e4\ub8e8\uc5c8\ub2e4.<\/p>\r\n\r\n<p>\uc608\uc81c 3: $42 \\le x \\le 50$\uc778 $S(x)$\uc758 \uac12\uc740 \uc21c\uc11c\ub300\ub85c $39, 39, 44, 45, 46, 47, 48, 49, 49$ \uc774\ubbc0\ub85c \ucd1d\ud569\uc778 406\uc774 \uc815\ub2f5\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc608\uc81c 4: \ub2f5\uc774 \ub9e4\uc6b0 \ucee4\uc9c8 \uc218 \uc788\uc73c\ubbc0\ub85c $10^9 + 7$\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud568\uc5d0 \uc720\uc758\ud558\uc790.<\/p>\r\n"},{"problem_id":"27928","problem_lang":"1","title":"Monotone Increasing Number","description":"<p>Albert likes to play with &quot;monotone increasing numbers.&quot; For an arbitrary non-negative number $x$, if the digits in $x$ from left-most (the highest place digit) to right-most (one&#39;s place digit) are never decreasing, then $x$ is said to be a monotone increasing number. For instance, 0, 5, 111, 234, and 2246 are monotone increasing numbers whereas 121 and 456 are not.<\/p>\r\n\r\n<p>Albert likes to play a game where, given an arbitrary integer $x$, he finds the largest monotone increasing number that does not exceed $x$ -- let us call such number $S(x)$ for convenience. According to this definition, we have $S(x) = x$ when $x$ is a monotone increasing number, and&nbsp;$S(5) = 5$ and $S(111) = 111$ are two examples. On the other hand, if $x$ is not a monotone increasing number then trivially $S(x) \\lt x$, and examples include $S(121) = 119$ and $S(465) = 459$.<\/p>\r\n\r\n<p>While watching how Albert plays this game, his sister suggested a more fun game:&nbsp;Given two integers $N \\le M$ that are no less than 0, find $F(N, M) = \\sum_{x=N}^{M} S(x)$. That is, $F(N, M)$ is the sum of all $S(x)$&#39;s for $x$ between $N$ and $M$, inclusive. Let&#39;s compute this value and help Albert and his sister.<\/p>\r\n","input":"<p>The first line of the input will contain $T$, the number of test cases.<\/p>\r\n\r\n<p>Each of the following $T$ lines will contain $N$ and $M$, separated by whitespace.<\/p>\r\n","output":"<p>For each test case, output $F(N, M)$ in each line. Since the answer may be too large, output the answer modulo $10^9 + 7$.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English","limit":"<ul>\r\n\t<li>$1 \\le T \\le 1000$<\/li>\r\n\t<li>$0&nbsp;\\le N \\le M \\le 10^{18}$<\/li>\r\n<\/ul>\r\n","sample_explain_1":"<p>Cases 1-2: Discussed in the problem statement.<\/p>\r\n\r\n<p>Case 3: For each $x$ with $42 \\le x \\le 50$, the values of $S(x)$ are: $39, 39, 44, 45, 46, 47, 48, 49, 49$. Hence, the answer is the sum of these values, 406.<\/p>\r\n\r\n<p>Case 4: Recall that you must output the answer modulo $10^9 + 7$.<\/p>\r\n"}]