Problem D
Blackboard Numbers
Your maths teacher, Professor Prime, likes writing integers on the blackboard at the end of class. You have been taking notes all semester, and have noticed two things:
-
There is an equal chance of the number being written in binary, octal, decimal, and hexadecimal.
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There seems to be no apparent connection between them.
Professor Prime has given you the task of finding the
connection between the numbers. He claims that there exists a
function $f$, such that on
day $i$, $f(i)$ would produce the number he
wrote on the blackboard. He seems to think that this task
should be relatively simple, given that he has been doing it
for a very long time, and you have a lot of numbers to plug
into the function.
You have decided not to complete his task, and rather calculate the probability that the number is a prime. You don’t remember why you decided to do so, but here we are, so I guess you just have to do it.
Input
The first line of the input consists of a single integer,
$T$, the number of test
cases.
Each of the $T$ test cases
consists of a single line with a string representing the number
he wrote that day.
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$1 \leq T \leq 200$
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Each number will consist of the characters 0-9 and A-F.
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Each number will have between 1 and 10 characters.
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Numbers in binary, octal, decimal and hexadecimal are numbers written in base 2, 8, 10 and 16, respectively. Hexadecimal numbers use extra digits A-F for 10-15.
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A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself.
Output
For each test case, output the probability that the number is a prime as a reduced fraction (that is, the greatest common divisor of the numerator and denominator is 1).
| Sample Input 1 | Sample Output 1 |
|---|---|
3 10 B 4 |
1/4 1/1 0/1 |
