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문제

정이면체군은 집합 D로, D상에서 곱셈 연산 *가 정의되어 있다. 집합 D는 두 원소 a와 b를 이용해 만들 수 있다. 곱셈 연산 *는 실제로 표기하는 경우는 거의 없고, 붙여서 쓰거나 제곱 형태로 나타낸다. 즉, a*a는 aa 또는 a2로 나타낸다. 또, a*a*b는 a2b1 또는 a2b로 나타낸다.

정이면체군 D는 두 파라미터 m과 n을 갖고, m ≥ 2, n ≥ 2이다. 이러한 정이면체군은 Dm,n으로 나타낸다. Dm,n사이의 관계는 다음과 같다.

am = a0 = 1, bn = b0 = 1, and ba = a(m-1)b.

따라서, Dm,n은 총 (mn)개의 원소를 가지며, 다음과 같다.

Dm,n = {ajbk | 0 ≤ j < m, 0 ≤ k < n} = {a0b0, a1b0, a2b0, ..., a(m-1)b0, ..., a(m-1)b(n-1)}.

예를 들어, m=6, n=2인 경우 D7,2는 다음과 같다.

{a0b0, a1b0, a2b0, a3b0, a4b0, a5b0, a6b0, a0b1, a1b1, a2b1, a3b1, a4b1, a5b1, a6b1}.

곱셉 연산은 교환 법칙이 성립하지 않기 때문에, ba ≠ ab이다. D7,2의 정의에 의하면 ba = a6b이다. 또, ajak = a(j+k)%m, bjbk = b(j+k)%n 이다. 따라서 다음과 같다.

(ajbk)(apbq) ≠ (a(j+p)%mb(k+q)%n)

D7,2의 두 원소 (a3b1)과 (a2b1)을 올바르게 곱하는 방법은 다음과 같다.

(a3b1)(a2b1) = (a3b0)(ba)(a1b1) = (a3b0)(a6b1)(a1b1)

ba = a6b이때문에 위와 같이 계산할 수 있다. 또, a3b0a6 = a3a6 = a(3+6)%7 = a2이기 때문에, (a3b0)(a6b1) = (a2b1)임을 알 수 있다. 따라서, 다음과 같이 계산할 수 있다.

(a3b0)(a6b1)(a1b1) = (a2b1)(a1b1).

이제, 위의 식을 아래와 같이 다시 계산할 수 있다.

(a2b1)(a1b1) = (a2b0)(ba)(a0b1) = (a2b0)(a6b1)(a0b1) = (a8b2) = (a1b0)

따라서, (a3b1)과 (a2b1)을 곱하면 (a3b1)(a2b1) = (a1b0)이 된다.

m과 n 그리고, 정이면체군 Dm,n 상의 두 원소 (ajbk)와 (apbq)가 주어졌을 때, 곱한 결과를 구하는 프로그램을 작성하시오.

입력

입력은 여러 개의 문제 세트로 이루어져 있다. 각 문제 세트는 입력 여러 줄로 이루어져 있다.

첫째 줄에는 문제ID와 m, n,p 가 주어지며, 공백으로 구분되어져 있다. m과 n은 1000을 넘지 않는 정수이고, p는 그 문제 세트에 있는 곱셈 문제의 개수이다. 다음 p개 줄에는 곱셈 문제가 주어지며, Dm,n 상의 두 원소가 주어진다. 각 원소는 a3b1형식이다. 각 문제 세트의 다음에는 바로 다음 문제 세트가 주어진다. 문제ID가 ZZ인 경우에는 입력의 끝을 나타낸다.

출력

각각의 곱셈 문제에 대해서, 다음을 출력한다.

ProblemID id: ajbk * apbq = arbs

id는 입력으로 주어진 문제ID, ajbk, apbq는 입력으로 주어진 두 원소, arbs는 결과이다.

예제 입력 1

C1 7 2 3
a3b1 a2b1
a5b1 a3b1
a2b1 a4b1
C2 9 3 2
a3b1 a2b1
a4b1 a3b1
ZZ 0 0 0

예제 출력 1

ProblemID C1: a3b1 * a2b1 = a1b0
ProblemID C1: a5b1 * a3b1 = a2b0
ProblemID C1: a2b1 * a4b1 = a5b0
ProblemID C2: a3b1 * a2b1 = a1b2
ProblemID C2: a4b1 * a3b1 = a1b2
[{"problem_id":"9736","problem_lang":"0","title":"\uc815\uc774\uba74\uccb4\uad70","description":"<p>\uc815\uc774\uba74\uccb4\uad70\uc740 \uc9d1\ud569 D\ub85c, D\uc0c1\uc5d0\uc11c \uacf1\uc148 \uc5f0\uc0b0 *\uac00 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4. \uc9d1\ud569 D\ub294 \ub450 \uc6d0\uc18c a\uc640 b\ub97c \uc774\uc6a9\ud574 \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uacf1\uc148 \uc5f0\uc0b0 *\ub294 \uc2e4\uc81c\ub85c \ud45c\uae30\ud558\ub294 \uacbd\uc6b0\ub294 \uac70\uc758 \uc5c6\uace0, \ubd99\uc5ec\uc11c \uc4f0\uac70\ub098 \uc81c\uacf1 \ud615\ud0dc\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989, a*a\ub294 aa \ub610\ub294 a<sup>2<\/sup>\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610, a*a*b\ub294 a<sup>2<\/sup>b<sup>1<\/sup> \ub610\ub294 a<sup>2<\/sup>b\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\r\n\r\n<p>\uc815\uc774\uba74\uccb4\uad70 D\ub294 \ub450 \ud30c\ub77c\ubbf8\ud130 m\uacfc n\uc744 \uac16\uace0, m &ge; 2, n &ge; 2\uc774\ub2e4. \uc774\ub7ec\ud55c \uc815\uc774\uba74\uccb4\uad70\uc740 D<sub>m,n<\/sub>\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. D<sub>m,n<\/sub>\uc0ac\uc774\uc758 \uad00\uacc4\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p>a<sup>m<\/sup> = a<sup>0<\/sup> = 1, b<sup>n<\/sup> = b<sup>0<\/sup> = 1, and ba = a<sup>(m-1)<\/sup>b.<\/p>\r\n\r\n<p>\ub530\ub77c\uc11c, D<sub>m,n<\/sub>\uc740 \ucd1d (mn)\uac1c\uc758 \uc6d0\uc18c\ub97c \uac00\uc9c0\uba70, \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p>D<sub>m,n<\/sub> = {a<sup>j<\/sup>bk | 0 &le; j &lt; m, 0 &le; k &lt; n} = {a<sup>0<\/sup>b<sup>0<\/sup>, a<sup>1<\/sup>b<sup>0<\/sup>, a<sup>2<\/sup>b<sup>0<\/sup>, ..., a<sup>(m-1)<\/sup>b<sup>0<\/sup>, ..., a<sup>(m-1)<\/sup>b<sup>(n-1)<\/sup>}.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, m=6, n=2\uc778 \uacbd\uc6b0 D<sub>7,2<\/sub>\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p>{a<sup>0<\/sup>b<sup>0<\/sup>, a<sup>1<\/sup>b<sup>0<\/sup>, a<sup>2<\/sup>b<sup>0<\/sup>, a<sup>3<\/sup>b<sup>0<\/sup>, a<sup>4<\/sup>b<sup>0<\/sup>, a<sup>5<\/sup>b<sup>0<\/sup>, a<sup>6<\/sup>b<sup>0<\/sup>, a<sup>0<\/sup>b<sup>1<\/sup>, a<sup>1<\/sup>b<sup>1<\/sup>, a<sup>2<\/sup>b<sup>1<\/sup>, a<sup>3<\/sup>b<sup>1<\/sup>, a<sup>4<\/sup>b<sup>1<\/sup>, a<sup>5<\/sup>b<sup>1<\/sup>, a<sup>6<\/sup>b<sup>1<\/sup>}.<\/p>\r\n\r\n<p>\uacf1\uc149 \uc5f0\uc0b0\uc740 \uad50\ud658 \ubc95\uce59\uc774 \uc131\ub9bd\ud558\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc5d0, ba &ne; ab\uc774\ub2e4. D<sub>7,2<\/sub>\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uba74 ba = a<sup>6<\/sup>b\uc774\ub2e4. \ub610, a<sup>j<\/sup>a<sup>k<\/sup> = a<sup>(j+k)%m<\/sup>, b<sup>j<\/sup>b<sup>k<\/sup> = b<sup>(j+k)%n<\/sup> \uc774\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p>(a<sup>j<\/sup>b<sup>k<\/sup>)(a<sup>p<\/sup>b<sup>q<\/sup>) &ne; (a<sup>(j+p)%m<\/sup>b<sup>(k+q)%n<\/sup>)<\/p>\r\n\r\n<p>D<sub>7,2<\/sub>\uc758 \ub450 \uc6d0\uc18c (a<sup>3<\/sup>b<sup>1<\/sup>)\uacfc (a<sup>2<\/sup>b<sup>1<\/sup>)\uc744 \uc62c\ubc14\ub974\uac8c \uacf1\ud558\ub294 \ubc29\ubc95\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p>(a<sup>3<\/sup>b<sup>1<\/sup>)(a<sup>2<\/sup>b<sup>1<\/sup>) = (a<sup>3<\/sup>b<sup>0<\/sup>)(ba)(a<sup>1<\/sup>b<sup>1<\/sup>) = (a<sup>3<\/sup>b<sup>0<\/sup>)(a<sup>6<\/sup>b<sup>1<\/sup>)(a<sup>1<\/sup>b<sup>1<\/sup>)<\/p>\r\n\r\n<p>ba = a<sup>6<\/sup>b\uc774\ub54c\ubb38\uc5d0 \uc704\uc640 \uac19\uc774 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \ub610, a<sup>3<\/sup>b<sup>0<\/sup>a<sup>6<\/sup> = a<sup>3<\/sup>a<sup>6<\/sup> = a<sup>(3+6)%7<\/sup> = a<sup>2<\/sup>\uc774\uae30 \ub54c\ubb38\uc5d0, (a<sup>3<\/sup>b<sup>0<\/sup>)(a<sup>6<\/sup>b<sup>1<\/sup>) = (a<sup>2<\/sup>b<sup>1<\/sup>)\uc784\uc744 \uc54c \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c, \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>(a<sup>3<\/sup>b<sup>0<\/sup>)(a<sup>6<\/sup>b<sup>1<\/sup>)(a<sup>1<\/sup>b<sup>1<\/sup>) = (a<sup>2<\/sup>b<sup>1<\/sup>)(a<sup>1<\/sup>b<sup>1<\/sup>).<\/p>\r\n\r\n<p>\uc774\uc81c, \uc704\uc758 \uc2dd\uc744 \uc544\ub798\uc640 \uac19\uc774 \ub2e4\uc2dc \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>(a<sup>2<\/sup>b<sup>1<\/sup>)(a<sup>1<\/sup>b<sup>1<\/sup>) = (a<sup>2<\/sup>b<sup>0<\/sup>)(ba)(a<sup>0<\/sup>b<sup>1<\/sup>) = (a<sup>2<\/sup>b<sup>0<\/sup>)(a<sup>6<\/sup>b<sup>1<\/sup>)(a<sup>0<\/sup>b<sup>1<\/sup>) = (a<sup>8<\/sup>b<sup>2<\/sup>) = (a<sup>1<\/sup>b<sup>0<\/sup>)<\/p>\r\n\r\n<p>\ub530\ub77c\uc11c, (a<sup>3<\/sup>b<sup>1<\/sup>)\uacfc (a<sup>2<\/sup>b<sup>1<\/sup>)\uc744 \uacf1\ud558\uba74 (a<sup>3<\/sup>b<sup>1<\/sup>)(a<sup>2<\/sup>b<sup>1<\/sup>) = (a<sup>1<\/sup>b<sup>0<\/sup>)\uc774 \ub41c\ub2e4.<\/p>\r\n\r\n<p>m\uacfc n \uadf8\ub9ac\uace0, \uc815\uc774\uba74\uccb4\uad70 D<sub>m,n<\/sub> \uc0c1\uc758 \ub450 \uc6d0\uc18c (a<sup>j<\/sup>b<sup>k<\/sup>)\uc640 (a<sup>p<\/sup>b<sup>q<\/sup>)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uacf1\ud55c \uacb0\uacfc\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ubb38\uc81c \uc138\ud2b8\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uac01 \ubb38\uc81c \uc138\ud2b8\ub294 \uc785\ub825 \uc5ec\ub7ec \uc904\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uccab\uc9f8 \uc904\uc5d0\ub294 \ubb38\uc81cID\uc640 m, n,p \uac00 \uc8fc\uc5b4\uc9c0\uba70, \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4\uc838 \uc788\ub2e4. m\uacfc n\uc740 1000\uc744 \ub118\uc9c0 \uc54a\ub294 \uc815\uc218\uc774\uace0, p\ub294 \uadf8 \ubb38\uc81c \uc138\ud2b8\uc5d0 \uc788\ub294 \uacf1\uc148 \ubb38\uc81c\uc758 \uac1c\uc218\uc774\ub2e4. \ub2e4\uc74c p\uac1c \uc904\uc5d0\ub294 \uacf1\uc148 \ubb38\uc81c\uac00 \uc8fc\uc5b4\uc9c0\uba70, D<sub>m,n<\/sub> \uc0c1\uc758 \ub450 \uc6d0\uc18c\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \uc6d0\uc18c\ub294 a3b1\ud615\uc2dd\uc774\ub2e4. \uac01 \ubb38\uc81c \uc138\ud2b8\uc758 \ub2e4\uc74c\uc5d0\ub294 \ubc14\ub85c \ub2e4\uc74c \ubb38\uc81c \uc138\ud2b8\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \ubb38\uc81cID\uac00 ZZ\uc778 \uacbd\uc6b0\uc5d0\ub294 \uc785\ub825\uc758 \ub05d\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\r\n","output":"<p>\uac01\uac01\uc758 \uacf1\uc148 \ubb38\uc81c\uc5d0 \ub300\ud574\uc11c, \ub2e4\uc74c\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>ProblemID id: ajbk * apbq = arbs<\/p>\r\n\r\n<p>id\ub294 \uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \ubb38\uc81cID, ajbk, apbq\ub294 \uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \ub450 \uc6d0\uc18c, arbs\ub294 \uacb0\uacfc\uc774\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"9736","problem_lang":"1","title":"Multiplication in a Dihedral Group","description":"<p>A dihedral group is a set D with a multiplication operation * defined on D. The set D is generated by two elements a and b. The multiplication operation * will not be explicitly written, so that a*a will be indicated by aa or by a<sup>2<\/sup>. Similarly, a*a*b will be indicated by a<sup>2<\/sup>b<sup>1<\/sup> or simply a<sup>2<\/sup>b.&nbsp;<\/p>\r\n\r\n<p>The dihedral group D has two integer parameters m and n, with m &ge; 2 and n &ge; 2, and so we write the dihedral group as D<sub>m,n<\/sub>. The relations that define D<sub>m,n<\/sub> are:&nbsp;<\/p>\r\n\r\n<p>a<sup>m<\/sup>&nbsp;= a<sup>0<\/sup>&nbsp;= 1, b<sup>n<\/sup>&nbsp;= b<sup>0<\/sup>&nbsp;= 1, and ba = a<sup>(m-1)<\/sup>b.<\/p>\r\n\r\n<p>Thus D<sub>m,n<\/sub> has (mn) elements, namely&nbsp;<\/p>\r\n\r\n<p>D<sub>m,n<\/sub>&nbsp;= {a<sup>j<\/sup>bk | 0 &le; j &lt; m, 0 &le; k &lt; n} = {a<sup>0<\/sup>b<sup>0<\/sup>, a<sup>1<\/sup>b<sup>0<\/sup>, a<sup>2<\/sup>b<sup>0<\/sup>, ..., a<sup>(m-1)<\/sup>b<sup>0<\/sup>, ..., a<sup>(m-1)<\/sup>b<sup>(n-1)<\/sup>}.<\/p>\r\n\r\n<p>For example, select m = 7 and n = 2. The elements of D<sub>7,2<\/sub> are<\/p>\r\n\r\n<p>{a<sup>0<\/sup>b<sup>0<\/sup>, a<sup>1<\/sup>b<sup>0<\/sup>, a<sup>2<\/sup>b<sup>0<\/sup>, a<sup>3<\/sup>b<sup>0<\/sup>, a<sup>4<\/sup>b<sup>0<\/sup>, a<sup>5<\/sup>b<sup>0<\/sup>, a<sup>6<\/sup>b<sup>0<\/sup>, a<sup>0<\/sup>b<sup>1<\/sup>, a<sup>1<\/sup>b<sup>1<\/sup>, a<sup>2<\/sup>b<sup>1<\/sup>, a<sup>3<\/sup>b<sup>1<\/sup>, a<sup>4<\/sup>b<sup>1<\/sup>, a<sup>5<\/sup>b<sup>1<\/sup>, a<sup>6<\/sup>b<sup>1<\/sup>}.<\/p>\r\n\r\n<p>Multiplication is not commutative, and so ba &ne; ab. In fact ba = a<sup>6<\/sup>b, according to the defining relations of D<sub>7,2<\/sub>. Also, a<sup>j<\/sup>a<sup>k<\/sup> = a<sup>(j+k)%m<\/sup>, and b<sup>j<\/sup>b<sup>k<\/sup> = b<sup>(j+k)%n<\/sup>. Thus&nbsp;<\/p>\r\n\r\n<p>(a<sup>j<\/sup>b<sup>k<\/sup>)(a<sup>p<\/sup>b<sup>q<\/sup>) &ne; (a<sup>(j+p)%m<\/sup>b<sup>(k+q)%n<\/sup>)<\/p>\r\n\r\n<p>To properly multiply the two elements (a<sup>3<\/sup>b<sup>1<\/sup>) and (a<sup>2<\/sup>b<sup>1<\/sup>) of D<sub>7,2<\/sub>, we proceed as follows,&nbsp;<\/p>\r\n\r\n<p>(a<sup>3<\/sup>b<sup>1<\/sup>)(a<sup>2<\/sup>b<sup>1<\/sup>) = (a<sup>3<\/sup>b<sup>0<\/sup>)(ba)(a<sup>1<\/sup>b<sup>1<\/sup>) = (a<sup>3<\/sup>b<sup>0<\/sup>)(a<sup>6<\/sup>b<sup>1<\/sup>)(a<sup>1<\/sup>b<sup>1<\/sup>)<\/p>\r\n\r\n<p>since ba = a<sup>6<\/sup>b.&nbsp;<\/p>\r\n\r\n<p>Also, since a<sup>3<\/sup>b<sup>0<\/sup>a<sup>6<\/sup>&nbsp;= a<sup>3<\/sup>a<sup>6<\/sup>&nbsp;= a<sup>(3+6)%7<\/sup>&nbsp;= a<sup>2<\/sup>, we can multiply the first two factors and get (a<sup>3<\/sup>b<sup>0<\/sup>)(a<sup>6<\/sup>b<sup>1<\/sup>) = (a<sup>2<\/sup>b<sup>1<\/sup>). Thus&nbsp;<\/p>\r\n\r\n<p>(a<sup>3<\/sup>b<sup>0<\/sup>)(a<sup>6<\/sup>b<sup>1<\/sup>)(a<sup>1<\/sup>b<sup>1<\/sup>) = (a<sup>2<\/sup>b<sup>1<\/sup>)(a<sup>1<\/sup>b<sup>1<\/sup>).<\/p>\r\n\r\n<p>Now we can rewrite this as follows,&nbsp;<\/p>\r\n\r\n<p>(a<sup>2<\/sup>b<sup>1<\/sup>)(a<sup>1<\/sup>b<sup>1<\/sup>) = (a<sup>2<\/sup>b<sup>0<\/sup>)(ba)(a<sup>0<\/sup>b<sup>1<\/sup>) = (a<sup>2<\/sup>b<sup>0<\/sup>)(a<sup>6<\/sup>b<sup>1<\/sup>)(a<sup>0<\/sup>b<sup>1<\/sup>) = (a<sup>8<\/sup>b<sup>2<\/sup>) = (a<sup>1<\/sup>b<sup>0<\/sup>)<\/p>\r\n\r\n<p>Thus, multiplying the two elements (a<sup>3<\/sup>b<sup>1<\/sup>) and (a<sup>2<\/sup>b<sup>1<\/sup>) gives,&nbsp;(a<sup>3<\/sup>b<sup>1<\/sup>)(a<sup>2<\/sup>b<sup>1<\/sup>) = (a<sup>1<\/sup>b<sup>0<\/sup>).<\/p>\r\n\r\n<p>Your problem is to write a computer program, given m, n, and any two elements (a<sup>j<\/sup>b<sup>k<\/sup>)&nbsp;and (a<sup>p<\/sup>b<sup>q<\/sup>) of the dihedral group D<sub>m,n<\/sub> that computes the correct product of the two.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","input":"<p>The input consists of several problem sets. Each problem set consists of several lines of input. The first line will contain the problemID, m, n, and p, separated by spaces. The integers m and n will not exceed 1000 in value. The number p is the number of multiplication problems for this problem set. The first line is followed by p lines of input, each line containing two elements of D<sub>m,n<\/sub> to multiply. Each element will be given in the form a3b1 instead of a<sup>3<\/sup>b<sup>1<\/sup>. Data for the next problem set will immediately follow the previous problem set. A set problemID of &ldquo;ZZ&rdquo; will indicate end of input.<\/p>\r\n","output":"<p>For each multiplication problem, print one line of the form<\/p>\r\n\r\n<p>ProblemID id: ajbk * apbq = arbs<\/p>\r\n\r\n<p>where id is the problemID given in the input, ajbk and apbq are the elements to multiply, and arbs is the correct answer.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]