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ZOJ Problem Set - 1665
Rational Irrationals

Time Limit: 2 Seconds      Memory Limit: 65536 KB

Rational numbers are numbers represented by ratios of two integers. For a prime number p, one of the elementary theorems in the number theory is that there is no rational number equal to the square root of p (abbreviated to SQRT(p)). Such numbers are called irrational numbers. It is also known that there are rational numbers arbitrarily close to SQRT(p).

Now, given a positive integer n, we define a set Qn of all rational numbers whose elements are represented by ratios of two positive integers both of which are less than or equal to n. For example, Q4 is a set of 11 rational numbers {1/1, 1/2, 1/3, 1/4, 2/1, 2/3, 3/1, 3/2, 3/4, 4/1, 4/3}. 2/2, 2/4, 3/3, 4/2 and 4/4 are not included here because they are equal to 1/1, 1/2, 1/1, 2/1 and 1/1, respectively.

Your job is to write a program that reads two integers p and n and reports two rational numbers x/y and u/v, where u/v < SQRT(p) < x/y and there are no other elements of Qn between u/v and x/y. When n is greater than SQRT(p), such a pair of rational numbers always exists.


The input consists of lines each of which contains two positive integers, a prime number p and an integer n in the following format.

p n

They are separated by a space character. You can assume that p and n are less than 10000, and that n is greater than SQRT(p). The end of the input is indicated by a line consisting of two zeros.


For each input line, your program should output a line consisting of the two rational numbers x/y and u/v (x/y > u/v) separated by a space character in the following format.

x/y u/v

They should be irreducible. For example, 6/14 and 15/3 are not accepted. They should be reduced to 3/7 and 5/1, respectively.

Sample Input

2 5
3 10
5 100
0 0

Sample Output

3/2 4/3
7/4 5/3
85/38 38/17

Source: Asia 1999, Kyoto (Japan)
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