from math import gcd
from lib_prime import primes
from lib_misc import modinv


def find_m(p: int) -> int:
    rs = []
    test_n = p * 1337
    for m in range(1, p):
        test_f = f(test_n, m)
        if test_f % p == 0:
            rs.append(m)
    assert len(rs) == 1
    return rs[0]


def calc_m(p: int) -> int:
    """
    (n // 10 + (n % 10) * m) % p == 0
    a + b * m == 0 (mod p)
        b * m == -a (mod p)
            m == -a * b**-1 (mod p)
    """
    n = p * 1337
    a = n // 10
    b = (n % 10)
    m = -a * modinv(b, p)
    m %= p
    return m


def f(n: int, m: int = 34) -> int:
    left = n // 10
    right = n % 10 * m
    r = left + right
    return r


def euler_274():
    ps = primes(10**7)
    sum_m = 0
    for p in ps:
        if gcd(p, 10) != 1:
            continue
        m = calc_m(p)
        if p < 1000:
            m_naiv = find_m(p)
            assert m == m_naiv
        sum_m += m
    return sum_m


if __name__ == "__main__":
    solution = euler_274()
    print(f"e274.py: {solution}")
    assert solution == 1601912348822