OC Practice Week 1

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2017-09-13 19:00 CEST

OC Practice Week 1

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2017-09-20 19:00 CEST
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Problem F
A Rational Sequence 2

A sequence of positive rational numbers is defined as follows:

An infinite full binary tree labeled by positive rational numbers is defined by:

  • The label of the root is $1/1$.

  • The left child of label $p/q$ is $p/(p+q)$.

  • The right child of label $p/q$ is $(p+q)/q$.

The top of the tree is shown in the following figure:

\includegraphics[]{f1.png}

The sequence is defined by doing a level order (breadth first) traversal of the tree (indicated by the light dashed line). So that:

\[ F(1) = 1/1, F(2) = 1/2, F(3) = 2/1, F(4) = 1/3, F(5) = 3/2, F(6) = 2/3, \ldots \]

Write a program which finds the value of $n$ for which $F(n)$ is $p/q$ for inputs $p$ and $q$.

Input

The first line of input contains a single integer $P$, $(1 \le P \le 1000)$, which is the number of data sets that follow. Each data set should be processed identically and independently. Each data set consists of a single line of input. It contains the data set number, $K$, a single space, the numerator, $p$, a forward slash (/) and the denominator, $q$, of the desired fraction.

Output

For each data set there is a single line of output. It contains the data set number, $K$, followed by a single space which is then followed by the value of $n$ for which $F(n)$ is $p/q$. Inputs will be chosen so $n$ will fit in a $32$-bit integer.

Sample Input 1 Sample Output 1
4
1 1/1
2 1/3
3 5/2
4 2178309/1346269
1 1
2 4
3 11
4 1431655765