from math import gcd
from fractions import Fraction


def product(xs):
    r = 1
    for x in xs:
        r *= x
    return r


def ratio(d):
    r = 0
    for i in range(1, d):
        if gcd(i, d) == 1:
            r += 1
    return Fraction(r, (d - 1))


def euler_243():
    target = Fraction(15499, 94744)  # 0.1635881955585578

    # The more prime factors, the lower the initial fraction gets. We figure
    # out empirically that using primes up to 23 yields results close to the
    # target fraction. We then iterate over the multiples to find the
    # solution.
    factors = [2, 3, 5, 7, 11, 13, 17, 19, 23]

    for mul in range(1, 10):
        d = mul * product(factors)
        r = ratio(d)
        if r < target:
            return d
    return None


if __name__ == "__main__":
    solution = euler_243()
    print("e243.py: " + str(solution))
    assert solution == 892371480