#!/usr/bin/python3

import math
from numpy import ndarray

def is_prime(num):
    if num < 4:
        return num == 2 or num == 3

    if num % 2 == 0 or num % 3 == 0:
        return 0

    limit = math.floor(math.sqrt(num)) + 1

    for i in range(5, limit, 6):
        if num % i == 0 or num % (i + 2) == 0:
            return 0

    return 1

def gcd(a, b):
    while b != 0:
        tmp = b
        b = a % b
        a = tmp

    return a

def lcm(a, b):
    return a * b / gcd(a, b)

def lcmm(values, n):
    if n == 2:
        return lcm(values[0], values[1])

    value = values[0]

    for i in range(1, n):
        return lcm(value, lcmm(values[i:], n-1))

def sieve(n):
    primes = ndarray((n,), int)

    primes[0] = 0
    primes[1] = 0
    primes[2] = 1
    primes[3] = 1

    for i in range(4, n, 2):
        primes[i] = 0
        primes[i+1] = 1

    limit = math.floor(math.sqrt(n))

    for i in range(3, limit, 2):
        if primes[i]:
            for j in range(i * i, n, 2 * i):
                primes[j] = 0

    return primes