PROOT - Primitive Root

In the field of Cryptography, prime numbers play an important role. We are interested in a scheme called "Diffie-Hellman" key exchange which allows two communicating parties to exchange a secret key. This method requires a prime number p and r which is a primitive root of p to be publicly known. For a prime number p, r is a primitive root if and only if it's exponents r, r2, r3 ... rp-1 are distinct (mod p).

Cryptography Experts Group (CEG) is trying to develop such a system. They want to have a list of prime numbers and their primitive roots. You are going to write a program to help them. Given a prime number p and another integer r < p, you need to tell whether r is a primitive root of p.

Input

There will be multiple test cases. Each test case starts with two integers p (p < 231) and n (1 ≤ n ≤ 100) separated by a space on a single line. p is the prime number we want to use and n is the number of candidates we need to check. Then n lines follow each containing a single integer to check. An empty line follows each test case and the end of test cases is indicated by p=0 and n=0 and it should not be processed. The number of test cases is at most 60.

Output

For each test case print "YES" (quotes for clarity) if r is a primitive root of p and "NO" (again quotes for clarity) otherwise.

Example

Input:
5 2
3
4

7 2
3
4

0 0

Output:
YES
NO
YES
NO

Explanation

In the first test case 31, 32 , 33 and 34 are respectively 3, 4, 2 and 1 (mod 5). So, 3 is a primitive root of 5.

41, 42 , 43 and 44 are respectively 4, 1, 4 and 1 respectively. So, 4 is not a primitive root of 5.


Added by:Swarnaprakash
Date:2009-01-14
Time limit:1s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ERL JS-RHINO NODEJS PERL6 VB.NET
Resource:Kurukshetra 09 OPC

hide comments
2015-06-23 13:11:56 Aditya Kumar
A helpful link : http://www.apfloat.org/prim.html
2013-06-23 16:40:15 Piyush Kumar
interesting Maths :)
2013-05-19 04:13:37 Ouditchya Sinha
Very nice problem, loved solving it. :)
2013-01-27 14:11:42 saket diwakar
i love such problems.....:)
2011-09-03 14:22:25 Romal Thoppilan
time limit should have been alleast 6s for non-mathematicians to solve it .
2011-07-14 17:01:06 Dark_night
good prob......
2011-06-21 17:46:36 The Champ
loved solving this one ;)
2010-04-13 18:30:58 :(){ :|: & };:

Seems like some demented maths Olympiad question with sufficiently huge number of test cases.
2009-12-26 15:08:03 Nitin Gangahar
Number theory :D
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