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문제

1801년에 칼 프리드리히 가우스 (1777-1855)는 현대 정수론의 토대를 마련한 "Disquisitiones Arithmeticae"를 발표했고, 오늘날까지 팔리고 있다. 이 책에서 가장 많이 나온 주제 중 하나는 제곱잉여이다.

소수 \(p\)와 정수 \(a \not\equiv 0 \pmod{p}\) 가 있다. \(a\)가 \(p\)에 대한 제곱잉여가 되려면 다음과 같은 조건을 만족하는 정수 \(x\)가 존재해야 한다.

\[x^2 \equiv a \pmod{p}\]

라그랑주 (1752-1833)은 아래와 같은 르장드르 기호를 만들었다.

\[\left(\frac{a}{p}\right) = 
\begin{cases}
 1 & a \text{ 가 } p \text{ 에 대한 제곱잉여인 경우} \\
-1 & a \text{ 가 } p \text{ 에 대한 제곱잉여가 아닌 경우} \\
\end{cases}\]

르장드르 기호를 계산하려면 다음과 같은 성질을 이용하면 좋으며, 서로 다른 홀수 소수 \(p\), \(q\)와 \(p\)로 나누어 떨어지지 않는 \(a\), \(b\)에 대해서만 성립한다.

  1. \(\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right)\)
  2. \(\left(\frac{1}{p}\right) = 1\)
  3. \(a \equiv b \pmod{p} \Rightarrow \left(\frac{a}{p}\right) = \left(\frac{b}{p}\right)\)
  4. \(\left(\frac{-1}{p}\right) = (-1)^{(p-1)/2}, \left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}\)
  5. \(\left(\frac{p}{q}\right)\left(\frac{q}{p}\right) = (-1)^{(p-1)(q-1)/4}\)

이제 다음과 같이 르장드르 기호를 쉽게 계산할 수 있다.

\[\left(\frac{29}{79}\right) = (-1)^{78\cdot28/4} \left(\frac{79}{29}\right)  = \left(\frac{79}{29}\right) = \left(\frac{-8}{29}\right) = \left(\frac{-1}{29}\right) \left(\frac{2}{29}\right)^{3} = \left(\frac{-1}{29}\right) \left(\frac{2}{29}\right) \\ = (-1)^{28/2}(-1)^{(29^2-1)/8} = (-1)^{14}(-1)^{105} = -1\]

입력

입력은 여러 개의 테스트 케이스로 이루어져 있고,각 테스트 케이스는 두 정수 \(a\)와 \(p\)로 이루어져 있다. \(p\) (\(2 < p < 10^9\))는 홀수 소수이고, \(a\)는 \(a \not\equiv 0 \pmod{p}\)와 \(\left| a \right| \le 10^9\)를 만족한다.

출력

각 테스트 케이스마다 "Scenario #i:"를 출력한다. i는 테스트 케이스 번호이며 1부터 시작한다. 그 다음 줄에 \(\left(\frac{a}{p}\right)\)를 출력하고 빈 줄을 하나 출력한다.

예제 입력 1

3
29 79
2 29
1 3

예제 출력 1

Scenario #1:
-1

Scenario #2:
-1

Scenario #3:
1
[{"problem_id":"7558","problem_lang":"0","title":"\uc81c\uacf1\uc789\uc5ec","description":"<p>1801\ub144\uc5d0 \uce7c \ud504\ub9ac\ub4dc\ub9ac\ud788 \uac00\uc6b0\uc2a4 (1777-1855)\ub294 \ud604\ub300 \uc815\uc218\ub860\uc758 \ud1a0\ub300\ub97c \ub9c8\ub828\ud55c &quot;Disquisitiones Arithmeticae&quot;\ub97c \ubc1c\ud45c\ud588\uace0, \uc624\ub298\ub0a0\uae4c\uc9c0 \ud314\ub9ac\uace0 \uc788\ub2e4. \uc774 \ucc45\uc5d0\uc11c \uac00\uc7a5 \ub9ce\uc774 \ub098\uc628 \uc8fc\uc81c \uc911 \ud558\ub098\ub294 \uc81c\uacf1\uc789\uc5ec\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc18c\uc218 \\(p\\)\uc640 \uc815\uc218 \\(a \\not\\equiv 0 \\pmod{p}\\) \uac00 \uc788\ub2e4. \\(a\\)\uac00 \\(p\\)\uc5d0 \ub300\ud55c \uc81c\uacf1\uc789\uc5ec\uac00 \ub418\ub824\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \uc815\uc218 \\(x\\)\uac00 \uc874\uc7ac\ud574\uc57c \ud55c\ub2e4.<\/p>\r\n\r\n<p>\\[x^2 \\equiv a \\pmod{p}\\]<\/p>\r\n\r\n<p>\ub77c\uadf8\ub791\uc8fc (1752-1833)\uc740 \uc544\ub798\uc640 \uac19\uc740 \ub974\uc7a5\ub4dc\ub974 \uae30\ud638\ub97c \ub9cc\ub4e4\uc5c8\ub2e4.<\/p>\r\n\r\n<p>\\[\\left(\\frac{a}{p}\\right) =&nbsp;<br \/>\r\n\\begin{cases}<br \/>\r\n&nbsp;1 &amp; a \\text{ \uac00 } p \\text{ \uc5d0 \ub300\ud55c \uc81c\uacf1\uc789\uc5ec\uc778 \uacbd\uc6b0} \\\\<br \/>\r\n-1 &amp; a \\text{ \uac00 } p \\text{ \uc5d0 \ub300\ud55c \uc81c\uacf1\uc789\uc5ec\uac00 \uc544\ub2cc \uacbd\uc6b0} \\\\<br \/>\r\n\\end{cases}\\]<\/p>\r\n\r\n<p>\ub974\uc7a5\ub4dc\ub974 \uae30\ud638\ub97c \uacc4\uc0b0\ud558\ub824\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uba74 \uc88b\uc73c\uba70, \uc11c\ub85c \ub2e4\ub978 \ud640\uc218 \uc18c\uc218 \\(p\\), \\(q\\)\uc640 \\(p\\)\ub85c \ub098\ub204\uc5b4 \ub5a8\uc5b4\uc9c0\uc9c0 \uc54a\ub294 \\(a\\), \\(b\\)\uc5d0 \ub300\ud574\uc11c\ub9cc \uc131\ub9bd\ud55c\ub2e4.<\/p>\r\n\r\n<ol>\r\n\t<li>\\(\\left(\\frac{ab}{p}\\right) = \\left(\\frac{a}{p}\\right)\\left(\\frac{b}{p}\\right)\\)<\/li>\r\n\t<li>\\(\\left(\\frac{1}{p}\\right) = 1\\)<\/li>\r\n\t<li>\\(a \\equiv b \\pmod{p} \\Rightarrow \\left(\\frac{a}{p}\\right) = \\left(\\frac{b}{p}\\right)\\)<\/li>\r\n\t<li>\\(\\left(\\frac{-1}{p}\\right) = (-1)^{(p-1)\/2}, \\left(\\frac{2}{p}\\right) = (-1)^{(p^2-1)\/8}\\)<\/li>\r\n\t<li>\\(\\left(\\frac{p}{q}\\right)\\left(\\frac{q}{p}\\right) = (-1)^{(p-1)(q-1)\/4}\\)<\/li>\r\n<\/ol>\r\n\r\n<p>\uc774\uc81c \ub2e4\uc74c\uacfc \uac19\uc774 \ub974\uc7a5\ub4dc\ub974 \uae30\ud638\ub97c \uc27d\uac8c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\\[\\left(\\frac{29}{79}\\right) = (-1)^{78\\cdot28\/4} \\left(\\frac{79}{29}\\right) &nbsp;= \\left(\\frac{79}{29}\\right) = \\left(\\frac{-8}{29}\\right) = \\left(\\frac{-1}{29}\\right) \\left(\\frac{2}{29}\\right)^{3} = \\left(\\frac{-1}{29}\\right) \\left(\\frac{2}{29}\\right) \\\\ = (-1)^{28\/2}(-1)^{(29^2-1)\/8} = (-1)^{14}(-1)^{105} = -1\\]<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uace0,\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ub450 \uc815\uc218 \\(a\\)\uc640 \\(p\\)\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \\(p\\) (\\(2 &lt; p &lt; 10^9\\))\ub294 \ud640\uc218 \uc18c\uc218\uc774\uace0, \\(a\\)\ub294 \\(a \\not\\equiv 0 \\pmod{p}\\)\uc640 \\(\\left| a \\right| \\le 10^9\\)\ub97c \ub9cc\uc871\ud55c\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 &quot;Scenario #i:&quot;\ub97c \ucd9c\ub825\ud55c\ub2e4. i\ub294 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4 \ubc88\ud638\uc774\uba70 1\ubd80\ud130 \uc2dc\uc791\ud55c\ub2e4. \uadf8 \ub2e4\uc74c \uc904\uc5d0 \\(\\left(\\frac{a}{p}\\right)\\)\ub97c \ucd9c\ub825\ud558\uace0 \ube48 \uc904\uc744 \ud558\ub098 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"7558","problem_lang":"1","title":"Quadratic Residues","description":"<p>In 1801, Carl Friedrich Gauss (1777-1855) published his &ldquo;Disquisitiones Arithmeticae&rdquo;, which basically created modern number theory and is still being sold today. One of the many topics treated in his book was the problem of quadratic residues.<\/p>\r\n\r\n<p>Consider a prime number \\(p\\) and an integer \\(a \\not\\equiv 0 \\pmod{p}\\). Then \\(a\\) is called a <em>quadratic residue mod p<\/em> if there is an integer \\(x\\) such that<\/p>\r\n\r\n<p>\\[x^2 \\equiv a \\pmod{p}\\]<\/p>\r\n\r\n<p>and a quadratic non residue otherwise. Lagrange (1752-1833) introduced the following notation, called the &ldquo;Legendre symbol&rdquo;:<\/p>\r\n\r\n<p>\\[\\left(\\frac{a}{p}\\right) =&nbsp;<br \/>\r\n\\begin{cases}<br \/>\r\n&nbsp;1 &amp; \\text{ if } a \\text{ is a quadratic residue mod } p \\\\<br \/>\r\n-1 &amp; \\text{ if } a \\text{ is a quadratic non residue mod } p \\\\<br \/>\r\n\\end{cases}\\]<\/p>\r\n\r\n<p>For the calculation of these symbol there are the following rules, valid only for distinct <strong>odd<\/strong> prime numbers \\(p\\), \\(q\\) and integers \\(a\\), \\(b\\) not divisible by \\(p\\):<\/p>\r\n\r\n<ol>\r\n\t<li>\\(\\left(\\frac{ab}{p}\\right) = \\left(\\frac{a}{p}\\right)\\left(\\frac{b}{p}\\right)\\)<\/li>\r\n\t<li>\\(\\left(\\frac{1}{p}\\right) = 1\\)<\/li>\r\n\t<li>\\(a \\equiv b \\pmod{p} \\Rightarrow \\left(\\frac{a}{p}\\right) = \\left(\\frac{b}{p}\\right)\\)<\/li>\r\n\t<li>\\(\\left(\\frac{-1}{p}\\right) = (-1)^{(p-1)\/2}, \\left(\\frac{2}{p}\\right) = (-1)^{(p^2-1)\/8}\\)<\/li>\r\n\t<li>\\(\\left(\\frac{p}{q}\\right)\\left(\\frac{q}{p}\\right) = (-1)^{(p-1)(q-1)\/4}\\)<\/li>\r\n<\/ol>\r\n\r\n<p>The statements 1. to 3. are obvious from the de\ufb01nition, 4. is called the Completion Theorem, and 5. is the famous Law of Quadratic Reciprocity for which Gauss himself gave no less than six different proofs in the &ldquo;Disquisitiones Arithmeticae&rdquo;. Knowing these facts, one can calculate all possible Legendre symbols as in the following example:<\/p>\r\n\r\n<p>\\[\\left(\\frac{29}{79}\\right) = (-1)^{78\\cdot28\/4} \\left(\\frac{79}{29}\\right) &nbsp;= \\left(\\frac{79}{29}\\right) = \\left(\\frac{-8}{79}\\right) = \\left(\\frac{-1}{29}\\right) \\left(\\frac{2}{29}\\right)^{3} = \\left(\\frac{-1}{29}\\right) \\left(\\frac{2}{29}\\right) \\\\ = (-1)^{28\/2}(-1)^{(29^2-1)\/8} = (-1)^{14}(-1)^{105} = -1\\]<\/p>\r\n","input":"<p>The first line contains the number of scenarios.&nbsp;<\/p>\r\n\r\n<p>For each scenario, there is one line containing the integers \\(a\\) and \\(p\\) separated by a single blank, where \\(2 &lt; p &lt; 10^9\\) is an odd prime, and \\(a\\) satis\ufb01es both &nbsp;\\(a \\not\\equiv 0 \\pmod{p}\\) and \\(\\left| a \\right| \\le 10^9\\).<\/p>\r\n","output":"<p>Start the output for every scenario with a line containing &ldquo;Scenario #i:&rdquo;, where i is the number of the scenario starting at 1. Then print a single line containing&nbsp;\\(\\left(\\frac{a}{p}\\right)\\)\u0011, followed by a blank line.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English"}]

출처

University > Tu-Darmstadt Programming Contest > TUD Contest 2003 7번

  • 문제를 번역한 사람: baekjoon
  • 문제의 오타를 찾은 사람: degurii