from math import gcd as bltin_gcd

import random
def primRoots(modulo):
    required_set = {num for num in range(1, modulo) if bltin_gcd(num, modulo) }
    return [g for g in range(1, modulo) if required_set == {pow(g, powers, modulo)
            for powers in range(1, modulo)}]

def ext_gcd(a,b):
    m, n = a, b
    xm, ym = 1, 0
    xn, yn = 0, 1
    while (n != 0):
        q = m // n # chia lấy phần nguyên+
        r = m % n # chia lấy phần dư
        xr, yr = xm - q*xn, ym - q*yn
        m = n
        xm, ym = xn, yn
        n = r
        xn, yn = xr, yr
    #return (xm, ym) # m = gcd(a,b) = xm * a + ym * b
    return ym if ym > 0 else ym + a

def powerMode(a, n, d):
  res=1
  r = a% d
  while(n > 0):
    if(n & 1): res = (res * r) % d
    r = (r*r) % d
    n>>=1
  return res

def rabinMiller(num):
    # Returns True if num is a prime number.

    s = num - 1
    t = 0
    while s % 2 == 0:
        # keep halving s while it is even (and use t
        # to count how many times we halve s)
        s = s // 2
        t += 1

    for trials in range(5): # try to falsify num's primality 5 times
        a = random.randrange(2, num - 1)
        v = pow(a, s, num)
        if v != 1: # this test does not apply if v is 1.
            i = 0
            while v != (num - 1):
                if i == t - 1:
                    return False
                else:
                    i = i + 1
                    v = (v ** 2) % num
    return True

def isPrime(num):
    if (num < 2):
        return False # 0, 1, and negative numbers are not prime

    lowPrimes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]

    if num in lowPrimes:
        return True

    for prime in lowPrimes:
        if (num % prime == 0):
            return False

    return rabinMiller(num)


