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문제

Alice와 Bob은 "소수 게임"을 즐겨한다.

소수 게임에서는 두 사람이 먼저 일정한 양의 정수 구간 (interval)을 먼저 고르고, 해당 구간에 속한 모든 정수 각각에 대해 미니 게임을 한 번씩 플레이 한다.

정확한 규칙은 아래와 같다.

  • 먼저, Alice가 두 개의 정수 A, k를 고른다 (이 때 2 ≤ k ≤ A-1 을 만족해야 한다).
    k는 미니 게임을 몇 번 할지를 결정하고, A는 정수 구간의 상한값을 결정한다.
  • 다음으로 Bob이 2 ≤ x ≤ A + 1 - k를 만족하는 정수 x를 고른다.
    x는 정수 구간의 하한값을 결정한다. 즉, x ≤ 정수구간 ≤ A가 된다. 
  • 두 사람은 이제 미니 게임을 k번 플레이 하는데, x이상 x+k-1 이하의 정수 각각에 대하여 아래 미니 게임을 플레이한다 (각각의 미니 게임에서 고른 정수를 N이라 하자).
    • 먼저, N을 종이에 적는다 (x ≤ N ≤ x+k-1). 그리고 Alice와 Bob이 번갈아 플레이 한다.
    • Alice가 먼저 시작하여 종이에 적힌 숫자보다 크지 않은 수 중 소수 (prime) P를 고른다.
    • 종이에 적힌 수가 X라면, 이를 지우고 X-P를 종이에 새로 적는다. 
    • Bob의 차례가 되어 이를 반복한다.
    • 만약 자신의 차례가 되었을 때 종이에 적힌 수가 0이나 1이라면 진다. (다시 말해, X-P를 종이에 적을 때 이 값이 0이나 1이면 이긴다.)

예를 들어 N = 8 이라면, Alice가 자신의 처음 차례에 7을 고르면 이긴다.

N = 10이라면 Alice가 자신의 차례에 고를 수 있는 소수는 2, 3, 5, 7 중 하나이다.

  • 2를 고른 경우, Bob이 자신의 차례에 받는 수는 8이고 이 때 7을 고르면 Bob이 이긴다.
  • 3을 고른 경우, Bob이 자신의 차례에 받는 수는 7이고, 이 때 7을 고르면 Bob이 이긴다.
  • 5를 고른 경우, Bob이 자신의 차례에 받는 수는 5이고, 이 때 5를 고르면 Bob이 이긴다.
  • 7을 고른 경우, Bob이 자신의 차례에 받는 수는 3이고, 이 때 3을 고르면 Bob이 이긴다.

따라서 N = 10인 경우, Bob이 최선을 다한다면 Alice가 이길 방법은 없다. (두 사람은 언제나 최선을 다해서 플레이 한다고 가정하자.)

Bob은 A, k가 주어졌을 때 x를 잘 골라서 자신의 승률을 최대화 하기로 했다. 만약 승률을 최대화 하는 x값이 여럿이라면 그 중 가장 작은 x를 찾기로 했다. Bob을 도와주자.

입력

첫 줄에 테스트 케이스의 수 T가 주어진다.

각 줄에 A와 k가 공백으로 구분되어 주어진다.

출력

각 테스트 케이스에 대해 두 개의 정수를 공백으로 구분하여 출력한다.

첫 수는 k번의 게임 중 Bob이 최대 몇 번을 이길 수 있는지 나타내고, 두 번째 수는 이를 위해 Bob이 선택해야하는 x값 중 최소 값을 나타낸다.

제한

  • 1 ≤ T ≤ 50
  • 3 ≤ A ≤ 100,000
  • 2 ≤ k ≤ A-1

예제 입력 1

5
12 2
12 3
12 4
100 11
1000 15

예제 출력 1

2 9
2 8
2 7
3 25
4 885
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(\ub2e4\uc2dc \ub9d0\ud574, X-P\ub97c \uc885\uc774\uc5d0 \uc801\uc744 \ub54c \uc774 \uac12\uc774 0\uc774\ub098 1\uc774\uba74 \uc774\uae34\ub2e4.)<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ul>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4 N = 8 \uc774\ub77c\uba74, Alice\uac00 \uc790\uc2e0\uc758 \ucc98\uc74c \ucc28\ub840\uc5d0 7\uc744 \uace0\ub974\uba74 \uc774\uae34\ub2e4.<\/p>\r\n\r\n<p>N = 10\uc774\ub77c\uba74 Alice\uac00 \uc790\uc2e0\uc758 \ucc28\ub840\uc5d0 \uace0\ub97c \uc218 \uc788\ub294 \uc18c\uc218\ub294 2, 3, 5, 7 \uc911 \ud558\ub098\uc774\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>2\ub97c \uace0\ub978 \uacbd\uc6b0, Bob\uc774 \uc790\uc2e0\uc758 \ucc28\ub840\uc5d0 \ubc1b\ub294 \uc218\ub294 8\uc774\uace0 \uc774 \ub54c 7\uc744 \uace0\ub974\uba74 Bob\uc774 \uc774\uae34\ub2e4.<\/li>\r\n\t<li>3\uc744 \uace0\ub978 \uacbd\uc6b0, Bob\uc774 \uc790\uc2e0\uc758 \ucc28\ub840\uc5d0 \ubc1b\ub294 \uc218\ub294 7\uc774\uace0, \uc774 \ub54c 7\uc744 \uace0\ub974\uba74 Bob\uc774 \uc774\uae34\ub2e4.<\/li>\r\n\t<li>5\ub97c \uace0\ub978 \uacbd\uc6b0, Bob\uc774 \uc790\uc2e0\uc758 \ucc28\ub840\uc5d0 \ubc1b\ub294 \uc218\ub294 5\uc774\uace0, \uc774 \ub54c 5\ub97c \uace0\ub974\uba74 Bob\uc774 \uc774\uae34\ub2e4.<\/li>\r\n\t<li>7\uc744 \uace0\ub978 \uacbd\uc6b0, Bob\uc774 \uc790\uc2e0\uc758 \ucc28\ub840\uc5d0 \ubc1b\ub294 \uc218\ub294 3\uc774\uace0, \uc774 \ub54c 3\uc744 \uace0\ub974\uba74 Bob\uc774 \uc774\uae34\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\ub530\ub77c\uc11c N = 10\uc778 \uacbd\uc6b0, Bob\uc774 \ucd5c\uc120\uc744 \ub2e4\ud55c\ub2e4\uba74 Alice\uac00 \uc774\uae38 \ubc29\ubc95\uc740 \uc5c6\ub2e4. (\ub450 \uc0ac\ub78c\uc740 \uc5b8\uc81c\ub098 \ucd5c\uc120\uc744 \ub2e4\ud574\uc11c \ud50c\ub808\uc774 \ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790.)<\/p>\r\n\r\n<p>Bob\uc740 A, k\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c x\ub97c \uc798 \uace8\ub77c\uc11c \uc790\uc2e0\uc758 \uc2b9\ub960\uc744 \ucd5c\ub300\ud654 \ud558\uae30\ub85c \ud588\ub2e4. \ub9cc\uc57d \uc2b9\ub960\uc744 \ucd5c\ub300\ud654 \ud558\ub294 x\uac12\uc774 \uc5ec\ub7ff\uc774\ub77c\uba74 \uadf8 \uc911 \uac00\uc7a5 \uc791\uc740 x\ub97c \ucc3e\uae30\ub85c \ud588\ub2e4. Bob\uc744 \ub3c4\uc640\uc8fc\uc790.<\/p>\r\n","input":"<p>\uccab \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc218 T\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uac01 \uc904\uc5d0 A\uc640 k\uac00 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574 \ub450 \uac1c\uc758 \uc815\uc218\ub97c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud558\uc5ec \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uccab \uc218\ub294 k\ubc88\uc758 \uac8c\uc784 \uc911 Bob\uc774 \ucd5c\ub300 \uba87 \ubc88\uc744 \uc774\uae38 \uc218 \uc788\ub294\uc9c0 \ub098\ud0c0\ub0b4\uace0, \ub450 \ubc88\uc9f8 \uc218\ub294 \uc774\ub97c \uc704\ud574 Bob\uc774 \uc120\ud0dd\ud574\uc57c\ud558\ub294 x\uac12 \uc911 \ucd5c\uc18c \uac12\uc744&nbsp;\ub098\ud0c0\ub0b8\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean","limit":"<ul>\r\n\t<li>1 &le; T &le; 50<\/li>\r\n\t<li>3 &le; A &le; 100,000<\/li>\r\n\t<li>2 &le; k &le; A-1<\/li>\r\n<\/ul>\r\n"},{"problem_id":"19595","problem_lang":"1","title":"Prime Number Game","description":"<p>Alice and Bob often play the &quot;Prime Number Game&quot;.<\/p>\r\n\r\n<p>In this game, the two players first pick an interval of positive integers, and then play a mini-game for each integer in this interval.<\/p>\r\n\r\n<p>Formal rules are defined as follows.<\/p>\r\n\r\n<ul>\r\n\t<li>First, Alice picks two integers A and k (2 &le; k &le; A-1).<br \/>\r\n\tk determines the number of mini games they will play, and A decides the maximum possible value of the said integer interval.<\/li>\r\n\t<li>Next, Bob pick an integer x with&nbsp;2 &le; x &le; A + 1 - k.<br \/>\r\n\tx determines the minimum possible value of the said integer interval. This ensures that x &le; integers in the interval &le; A.<\/li>\r\n\t<li>Now the two players will play a mini-game exactly k times, once for each integer between x and x+k-1, inclusive (let N be the integer picked for a specific mini-game).\r\n\t<ul>\r\n\t\t<li>First, write down &quot;N&quot; on a piece of paper (x &le; N &le; x+k-1). Alice and Bob will take turns to play this mini-game.<\/li>\r\n\t\t<li>Alice goes first, and picks a prime number P that is no greater than the number written on the paper.<\/li>\r\n\t\t<li>If X is the number written on the paper, then Alice erases X, and writes X-P.<\/li>\r\n\t\t<li>Bob goes next, and repeat this.<\/li>\r\n\t\t<li>If a player begins his\/her turn with 0 or 1 written on the paper, the player loses. (In other words, if a player writes 0 or 1 on the paper, the player wins.)<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ul>\r\n\r\n<p>For instance, when N = 8, Alice can pick P = 7 in turn 1, and wins.<\/p>\r\n\r\n<p>When N = 10, Alice can pick any of 2, 3, 5, or 7.<\/p>\r\n\r\n<ul>\r\n\t<li>If Alice picks 2, then Bob starts his turn with 8, and Bob can choose 7 to win.<\/li>\r\n\t<li>If Alice picks 3, then Bob starts his turn with 7, and Bob can choose 7 to win.<\/li>\r\n\t<li>If Alice picks 5, then Bob starts his turn with 5, and Bob can choose 5 to win.<\/li>\r\n\t<li>If Alice picks 7, then Bob starts his turn with 3, and Bob can choose 3 to win.<\/li>\r\n<\/ul>\r\n\r\n<p>Hence, when N = 10, if Bob plays optimally, then Alice can&#39;t win.&nbsp;<br \/>\r\nIn this problem, we&#39;ll assume that both players will always play optimally.&nbsp;<\/p>\r\n\r\n<p>Bob wants to maximize his win rate&nbsp;by choosing x carefully, given that Alice has picked A and k.&nbsp;<br \/>\r\nIf there are many such x&#39;s, then Bob wants to find the smallest such x.<\/p>\r\n","input":"<p>The first line will contain the number of test cases, T.<\/p>\r\n\r\n<p>Each test case will contain A and k separated by a whitespace.<\/p>\r\n","output":"<p>For each test case, you must output two integers separated by a whitespace.<\/p>\r\n\r\n<p>The first number is the maximum number of mini-games Bob can win, and the second number is the smallest value of x to accomplish that.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English","limit":"<ul>\r\n\t<li>1 &le; T &le; 50<\/li>\r\n\t<li>3 &le; A &le; 100,000<\/li>\r\n\t<li>2 &le; k &le; A-1<\/li>\r\n<\/ul>\r\n"}]