Problem:


Score : 800 points

Problem Statement

You are given a polynomial of degree N with integer coefficients: f(x)=a_Nx^N+a_{N-1}x^{N-1}+...+a_0. Find all prime numbers p that divide f(x) for every integer x.

Constraints

0 \leq N \leq 10^4

|a_i| \leq 10^9(0\leq i\leq N)

a_N \neq 0

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_N
:
a_0

Output

Print all prime numbers p that divide f(x) for every integer x, in ascending order.

Sample Input 1

2
7
-7
14

Sample Output 1

2
7

2 and 7 divide, for example, f(1)=14 and f(2)=28.

Sample Input 2

3
1
4
1
5

Sample Output 2

There may be no integers that satisfy the condition.

Sample Input 3

0
998244353

Sample Output 3

998244353



Sample input:
2
7
-7
14


Sample output:
2
7

Submission status: Runtime Error

Solution code:
import sys
from fractions import gcd


def prime(n):
    ret = []
    for i in range(2, int(n ** 0.5) + 1):
        if n % i == 0:
            ret.append(i)
            while n % i == 0:
                n = n // i
        if n == 1:
            break
    if n > 1:
        ret.append(n)
    return ret


def solve(n, aaa):
    if n == 0:
        return prime(aaa[0])

    g = aaa[0]
    for a in aaa:
        g = gcd(g, a)
    ret = prime(g)
    if 2 not in ret and n % 2 == 0 and aaa[-1] % 2 == 0:
        ret = [2] + ret
    return ret


n = int(input())
aaa = list(map(int, sys.stdin))
ans = solve(n, aaa)
print('\n'.join(map(str, ans)))

Submission status: Runtime Error

Solution code:
import math
import time
import random

t1 = time.time() 

#euclid
A=[2*k+1 for k in range(1,15813)]
p=[2]

while(1):
    y=A.pop(0)
    if y > math.sqrt(15813):
        break
    p.append(y)
    A=[x for x in A if x%y!=0]
p=p+A
N=int(input())
X=[]
for i in range(N+1):
    X.append(int(input()))
#cands
if X[-1]!=0:
    C=[x for x in p if X[-1]%x==0]
if len(C)==0 and sum([ X[-1]==x for x in [-1, 0, 1] ])==0 :
    C=[abs(X[-1])]
    
tmax=1.5
kflag=0
Q=0
while(len(C)!=0):
    Q+=1
    Cnew=[]
    z=random.randint(1,max(C))
    for c in C:
        if kflag:
            break
        S=0
        t2=time.time()
        if ( t2-t1 ) > tmax*0.9:
            kflag=1
        tmp=1
        for i in range(N,-1,-1):
            S +=  (X[i] * tmp)%c
            tmp=(tmp*z)%c

        #print(S,S%c,c)
        if S%c==0:
            Cnew.append(c)
    if kflag:
        break
    else:
        C=Cnew[:]
for c in sorted(C):
    print(c)

Submission status: Runtime Error

Solution code:
N=int(input())
a=[int(input()) for i in range(N+1)][::-1]
def check(P):
    b=[a[i]%P for i in range(1,N+1)]
    if N<P:
        for i in b:
            if i!=0:
                return False
        return True
    else:
        c=[0 for i in range(P-1)]
        for i in range(N):
            c[i%(P-1)]+=b[i]
            c[i%(P-1)]%=P
        for i in c:
            if i!=0:
                return False
        return True
Plist=[]
X=[1 for i in range(N+1)]
X[0]=0;X[1]=0
for i in range(2,N+1):
    for j in range(i,N+1,i):
        if i==j:
            if X[i]==0:
                break
        else:
            X[j]=0
for j in range(N+1):
    if X[j]==1:
        if check(j):
            print(j)

Submission status: Runtime Error

Solution code:
import sys
sys.setrecursionlimit(10**5)

def prime_decomposition(n):
	prime = {}
	p_max = int(n**0.5)+1
	for p in range(2, p_max+1):
		if n%p == 0:
			prime[p] = 0
			while n%p == 0:
				n = n//p
				prime[p] += 1
	if n!=1:
		prime[n] = 1

	return prime

def func_gcd(x,y):
	x = abs(x)
	y = abs(y)
	if y>x:
		tmp = x
		x = y
		y = tmp
	if x % y == 0:
		return y
	x %= y
	return func_gcd(y,x)

n = int(input())
a = [int(input()) for _ in range(n+1)]

gcd = a[0]
for ai in a:
	gcd = func_gcd(gcd, ai)

# primes = prime_decomposition(gcd)


# if 2 not in primes.keys():
# 	if a[-1]%2 == 0 and sum(a)%2 == 0:
# 		print(2)
# for k in primes.keys():
# 	print(k)

Submission status: Runtime Error

Solution code:
import sys
sys.setrecursionlimit(10**5)

def prime_decomposition(n):
	prime = {}
	p_max = int(n**0.5)+1
	for p in range(2, p_max+1):
		if n%p == 0:
			prime[p] = 0
			while n%p == 0:
				n = n//p
				prime[p] += 1
	if n!=1:
		prime[n] = 1

	return prime

def func_gcd(x,y):
	x = abs(x)
	y = abs(y)
	if y>x:
		tmp = x
		x = y
		y = tmp
	if x % y == 0:
		return y
	x %= y
	return func_gcd(y,x)

n = int(input())
a = [int(input()) for _ in range(n+1)]

gcd = a[0]
for ai in a:
	gcd = func_gcd(gcd, ai)

primes = prime_decomposition(gcd)


if 2 not in primes.keys():
	if a[-1]%2 == 0 and sum(a)%2 == 0:
		print(2)
for k in primes.keys():
	print(k)

Submission status: Runtime Error

Solution code:
def prime_decomposition(n):
	prime = {}
	p_max = int(n**0.5)+1
	for p in range(2, p_max+1):
		if n%p == 0:
			prime[p] = 0
			while n%p == 0:
				n = n//p
				prime[p] += 1
	if n!=1:
		prime[n] = 1

	return prime

def func_gcd(x,y):
	x = abs(x)
	y = abs(y)
	if y>x:
		tmp = x
		x = y
		y = tmp
	if x % y == 0:
		return y
	x %= y
	return func_gcd(y,x)

n = int(input())
a = [int(input()) for _ in range(n+1)]

gcd = a[0]
for ai in a:
	gcd = func_gcd(gcd, ai)

Submission status: Runtime Error

Solution code:
def prime_decomposition(n):
	prime = {}
	p_max = int(n**0.5)+1
	for p in range(2, p_max+1):
		if n%p == 0:
			prime[p] = 0
			while n%p == 0:
				n = n//p
				prime[p] += 1
	if n!=1:
		prime[n] = 1

	return prime

def func_gcd(x,y):
	x = abs(x)
	y = abs(y)
	if y>x:
		tmp = x
		x = y
		y = tmp
	if x % y == 0:
		return y

	x %= y
	return func_gcd(y,x)

n = int(input())
a = [int(input()) for _ in range(n+1)]

gcd = a[0]
for ai in a:
	gcd = func_gcd(gcd, ai)

# primes = prime_decomposition(gcd)


# if 2 not in primes.keys():
# 	if a[-1]%2 == 0 and sum(a)%2 == 0:
# 		print(2)
# for k in primes.keys():
# 	print(k)

Submission status: Runtime Error

Solution code:
#!usr/bin/env python3
from collections import defaultdict
from collections import deque
from heapq import heappush, heappop
import sys
import math
import bisect
import random
def LI(): return list(map(int, sys.stdin.readline().split()))
def I(): return int(sys.stdin.readline())
def LS():return list(map(list, sys.stdin.readline().split()))
def S(): return list(sys.stdin.readline())[:-1]
def IR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = I()
    return l
def LIR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LI()
    return l
def SR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = S()
    return l
def LSR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LS()
    return l
sys.setrecursionlimit(1000000)
mod = 998244353

#C
def C():
    n = I()
    s = S()
    ans = float("inf")
    b = [0 for i in range(n)]
    for i in range(n):
        if s[i] == "#":
            b[i] = 1
    w = [0 for i in range(n)]
    for i in range(n):
        if s[i] == ".":
            w[i] = 1
    b.insert(0,0)
    w.insert(0,0)
    for i in range(n):
        b[i+1] += b[i]
        w[i+1] += w[i]
    for i in range(n+1):
        m = b[i]+w[n]-w[i]
        ans = min(ans,m)
    print(ans)
    return

#D
def D():
    n = I()
    return

#E
def E():
    def factorize(n):
        if n == 1:
            return []
        if n < 4:
            return [n]
        i = 2
        res = []
        m = n
        while i**2 <= n:
            if m%i == 0:
                while m%i == 0:
                    m //= i
                res.append(i)
            i += 1
        if m != 1:
            res.append(m)
        return res
    def f(x,m):
        res = a[-1]%m
        for i in range(n):
            res += a[i]*pow(x,n-i,m)%m
            if res >= m:
                res %= m
        return res
    def gcd(a,b):
        if a == 0:
            return b
        return gcd(b%a,a)
    n = I()
    a = IR(n+1)
    ans = [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, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973]
    j = 0
    while j < len(ans):
        b = [a[i]%ans[j] for i in range(n+1)]
        for i in range(n-ans[j]+1):
            b[i+ans[j]-1] -= b[i]
            b[i+ans[j]-1] %= ans[j]
            b[i] = 0
        if sum(b) != 0:
            ans.pop(j)
            j -= 1
        j += 1
    g = abs(a[0])
    l = abs(a[0])
    for i in a[1:]:
        if i == 0:
            g = 0
            l = 0
            break
        g = gcd(g,abs(i))
        l *= abs(i)//g
    k = factorize(l)
    j = 0
    while j < len(k):
        b = [a[i]%k[j] for i in range(n+1)]
        for i in range(n-k[j]+1):
            b[i+k[j]-1] -= b[i]
            b[i+k[j]-1] %= k[j]
            b[i] = 0
        if sum(b) != 0:
            k.pop(j)
            j -= 1
        j += 1
    ans += k
    ans = list(set(ans))
    ans.sort()
    for i in ans:
        print(i)
    return

#F
def F():
    return


#Solve
if __name__ == "__main__":
    E()

Submission status: Runtime Error

Solution code:
def prime_decomposition(n):
	prime = {}
	p_max = int(n**0.5)+1
	for p in range(2, p_max+1):
		if n%p == 0:
			prime[p] = 0
			while n%p == 0:
				n = n//p
				prime[p] += 1
	if n!=1:
		prime[n] = 1

	return prime

def func_gcd(x,y):
	x = abs(x)
	y = abs(y)
	if y>x:
		tmp = x
		x = y
		y = tmp
	if x % y == 0:
		return y

	x %= y
	return func_gcd(y,x)

n = int(input())
a = [int(input()) for _ in range(n+1)]

gcd = a[0]
for ai in a:
	gcd = func_gcd(gcd, ai)

primes = prime_decomposition(gcd)


if 2 not in primes.keys():
	if a[-1]%2 == 0 and sum(a)%2 == 0:
		print(2)
for k in primes.keys():
	print(k)

Submission status: Runtime Error

Solution code:
import math
import copy


def trial_division(n):
    factor = []
    tmp = int(math.sqrt(n)) + 1
    for num in range(2, tmp):
        while n % num == 0:
            n //= num
            factor.append(num)

    if not factor:
        return set([n])
    else:
        factor.append(n)
        return set(factor)


def solve():
    n = int(input())
    a = []
    for i in range(n + 1):
        a.append(int(input()))

    primes = trial_division(a[-1])

    for e in a[:-1]:
        output = copy.deepcopy(primes)
        for p in primes:
            if p % e != 0:
                output.remove(p)
        primes = output
    for p in primes:
        print(p)


if __name__ == '__main__':
    solve()

Submission status: Runtime Error

Solution code:
import math
from sympy.ntheory import factorint

# def prime_decomposition(n):
#     p = 2
#     prime = {}
#     p_max = n**0.5
#     while n!=1 and p<=n**0.5:
#         while n%p == 0:
#             n = n/p
#             if p not in prime:
#                 prime[p] = 0
#             prime[p] += 1
#         p += 1
#     return prime

def func_gcd(x,y):
	x = abs(x)
	y = abs(y)
	if y>x:
		tmp = x
		x = y
		y = tmp
	if x % y == 0:
		return y

	x %= y
	return func_gcd(y,x)

n = int(input())
a = [int(input()) for _ in range(n+1)]

gcd = a[0]
for ai in a:
	gcd = func_gcd(gcd, ai)

primes = factorint(gcd)


if 2 not in primes.keys():
	if a[-1]%2 == 0 and sum(a)%2 == 0:
		print(2)
for k in primes.keys():
	print(k)

Submission status: Runtime Error

Solution code:
import math
from sympy.ntheory import factorint

# def prime_decomposition(n):
#     p = 2
#     prime = {}
#     p_max = n**0.5
#     while n!=1 and p<=n**0.5:
#         while n%p == 0:
#             n = n/p
#             if p not in prime:
#                 prime[p] = 0
#             prime[p] += 1
#         p += 1
#     return prime

def func_gcd(x,y):
	x = abs(x)
	y = abs(y)
	if y>x:
		tmp = x
		x = y
		y = tmp
	if x % y == 0:
		return y

	x %= y
	return func_gcd(y,x)

	


n = int(input())
a = [int(input()) for _ in range(n+1)]

gcd = a[0]
for i in range(n+1):
	gcd = func_gcd(gcd, a[i])

primes = factorint(gcd)


if 2 not in primes.keys():
	if a[-1]%2 == 0 and sum(a)%2 == 0:
		print(2)
for k in primes.keys():
	print(k)

Submission status: Runtime Error

Solution code:
inpl = lambda: list(map(int,input().split()))

N = int(input())
a = []
for i in range(N):
    a.append(int(input()))
from functools import reduce
import fraction
#import math as fraction
def gcd_list(numbers):
    return reduce(fraction.gcd, numbers)

g = gcd_list(a)
plist = []

while i*i <= g:
    if g % i == 0:
        while g % i == 0:
            g %= i
    i += 1
if g > 1:
    plist.append(g)

for p in plist:
    print(p)

Submission status: Runtime Error

Solution code:
N=int(input())
t=[int(input()) for i in range(N+1)]
a=[t[-i-1] for i in range(N+1)]

def f(x):
    v=a[0]
    y=x
    for i in range(N):
        v+=a[i]*y
        y*=x
    return v

def primedecomp(n):
    prime_dic={}
    for i in range(2,n):
        if i**2>n:
            break
        if n%i==0:
            prime_dic[i]=0
        while n%i==0:
            n//=i
            prime_dic[i]+=1

    if n>1:
        prime_dic[n]=1
    return prime_dic

def lcm(a,b):
    if a<b:
        a,b=b,a
    if a%b==0:
        return b
    else:
        return lcm(b,a%b)
def check(p):
    for i in range(1,p+1):
        if f(i)%p!=0:
            return False
    return True
l=0
for i in range(N+1):
    if l==0 and a[i]!=0:
        l=a[i]
    elif a[i]!=0:
        l=lcm(l,a[i])
v=0
x=0
if a[0]==0:
    ps=list(primedecomp(min(abs(f(1)),f(-1))).keys())
else:
    ps=list(primedecomp(a[0]).keys())

ans=[]
for p in ps:
    if l%p==0:
        ans.append(p)
    elif check(p):
        ans.append(p)
for p in ans:
    print(p)

Submission status: Runtime Error

Solution code:
import math
from sympy.ntheory import factorint

# def prime_decomposition(n):
#     p = 2
#     prime = {}
#     p_max = n**0.5
#     while n!=1 and p<=n**0.5:
#         while n%p == 0:
#             n = n/p
#             if p not in prime:
#                 prime[p] = 0
#             prime[p] += 1
#         p += 1
#     return prime

def func_gcd(x,y):
	if y>x:
		tmp = x
		x = y
		y = tmp
	x %= y
	return func_gcd(y,x)

	


n = int(input())
a = [int(input()) for _ in range(n+1)]

gcd = a[0]
for i in range(n+1):
	gcd = func_gcd(gcd, a[i])

primes = factorint(gcd)


if 2 not in primes.keys():
	if a[-1]%2 == 0 and sum(a)%2 == 0:
		print(2)
for k in primes.keys():
	print(k)

Submission status: Runtime Error

Solution code:
import math
from sympy.ntheory import factorint

# def prime_decomposition(n):
#     p = 2
#     prime = {}
#     p_max = n**0.5
#     while n!=1 and p<=n**0.5:
#         while n%p == 0:
#             n = n/p
#             if p not in prime:
#                 prime[p] = 0
#             prime[p] += 1
#         p += 1
#     return prime
	


n = int(input())
a = [int(input()) for _ in range(n+1)]

gcd = a[0]
for i in range(n+1):
	gcd = math.gcd(gcd, a[i])

primes = factorint(gcd)


if 2 not in primes.keys():
	if a[-1]%2 == 0 and sum(a)%2 == 0:
		print(2)
for k in primes.keys():
	print(k)

Submission status: Runtime Error

Solution code:
N=int(input())
t=[int(input()) for i in range(N+1)]
a=[t[-i-1] for i in range(N+1)]

def f(x):
    v=a[0]
    y=x
    for i in range(N):
        v+=a[i]*y
        y*=x
    return v

def primedecomp(n):
    prime_dic={}
    for i in range(2,n):
        if i**2>n:
            break
        if n%i==0:
            prime_dic[i]=0
        while n%i==0:
            n//=i
            prime_dic[i]+=1

    if n>1:
        prime_dic[n]=1
    return prime_dic

def lcm(a,b):
    if a<b:
        a,b=b,a
    if a%b==0:
        return b
    else:
        return lcm(b,a%b)
def check(p):
    for i in range(1,p+1):
        if f(i)%p!=0:
            return False
    return True
l=0
for i in range(N+1):
    if l==0 and a[i]!=0:
        l=a[i]
    elif a[i]!=0:
        l=lcm(l,a[i])
v=0
x=0
if a[0]==0:
    ps=list(primedecomp(min(abs(f(1)),f(-1))).keys())
else:
    ps=list(primedecomp(a[0]).keys())
    
from collections import defaultdict
for i in range(N):
    if a[i+1]
    l=lcm(l,a[i+1])
ans=[]
for p in ps:
    if l%p==0:
        ans.append(p)
    elif check(p):
        ans.append(p)
for p in ans:
    print(p)

Submission status: Runtime Error

Solution code:
import math

n=int(input())
l=[int(input()) for i in range(n+1)]
def factor(m):
    num=m
    res=[]
    i=2
    while i*i<=m:
        if num%i:
            i+=1
        else:
            res.append(i)
            num//=i
    if num!=1:
        res.append(num)
    return res
res=list(set(factor(l[-1])))
#print(res)

def primes(n):
    is_prime = [True] * (n + 1)
    is_prime[0] = False
    is_prime[1] = False
    for i in range(2, int(n**0.5) + 1):
        if not is_prime[i]:
            continue
        for j in range(i * 2, n + 1, i):
            is_prime[j] = False
    return [i for i in range(n + 1) if is_prime[i]]
    
prime_lis=primes(10**4)
#print(prime_lis)
s=[]
for i in prime_lis:
    cnt=0
    for j in range(n+1):
        cnt+=l[n-j]*i**j
    s.append(cnt)
res.sort()
for i in res:
    check=True
    for j in s:
        if j%i!=0:
            check=False
            break
    if check:
        print(i)

Submission status: Runtime Error

Solution code:
# -*- coding: utf-8 -*-
import numpy as np

A =list()
N =int(raw_input())
for i in range(N):
    a =int(raw_input())
    A =np.append(A,a)

def primes(n):
    is_prime = [True] * (n + 1)
    is_prime[0] = False
    is_prime[1] = False
    for i in range(2, int(n**0.5) + 1):
        if not is_prime[i]:
            continue
        for j in range(i * 2, n + 1, i):
            is_prime[j] = False
    return [i for i in range(n + 1) if is_prime[i]]
def Y(n):
    y = 0
    for i in range(N):
        y = y + n**i*A[N-1-i]
    return y

so = list(primes(int(A[N-1])))

for n in range(20):
    for i in so:
        Y = Y(n)
        if Y%i != 0
            so.remove(i)
for n in so:
    print so

Submission status: Runtime Error

Solution code:
import math

n=int(input())
l=[int(input()) for i in range(n+1)]
def factor(m):
    num=m
    res=[]
    i=2
    while i*i<=m:
        if num%i:
            i+=1
        else:
            res.append(i)
            num//=i
    if num!=1:
        res.append(num)
    return res
res=list(set(factor(l[-1])))
#print(res)

def primes(n):
    is_prime = [True] * (n + 1)
    is_prime[0] = False
    is_prime[1] = False
    for i in range(2, int(n**0.5) + 1):
        if not is_prime[i]:
            continue
        for j in range(i * 2, n + 1, i):
            is_prime[j] = False
    return [i for i in range(n + 1) if is_prime[i]]
    
prime_lis=primes(math.ceil(math.sqrt(l[-1])))
#print(prime_lis)
s=[]
for i in prime_lis:
    cnt=0
    for j in range(n+1):
        cnt+=l[n-j]*i**j
    s.append(cnt)
res.sort()
for i in res:
    check=True
    for j in s:
        if j%i!=0:
            check=False
            break
    if check:
        print(i)

Submission status: Runtime Error

Solution code:
#coding: utf-8

import math

def yakusuu(n):
    n = abs(n)
    nums = []
    k = 2
    maximum = math.sqrt(n) + 1
    while n > 1 and k <= maximum:
        flag = False
        while n%k == 0:
            n = int(n / k)
            flag = True
        if flag:
            nums.append(k)
            nums.append(n)
        k += 1
    nums.append(n)
    return nums

def yakusuu2(n):
    nums = [n]
    k = 2
    maximum = math.sqrt(n) + 1
    while k <= maximum:
        if (n % k) == 0:
            nums.append(k)
            nums.append(int(n / k))
        k += 1
    return nums



N = int(input())
nums = [int(input()) for i in range(0, N+1)]
nums.reverse()
f0 = nums[0]

for y in yakusuu(f0):
    for x in [1, 2, 3, -1]:
        powerdX = 1
        res = 0
        for n in nums:
            res += powerdX * n
            powerdX = powerdX * x
        if res % y != 0:
            break
    else:
        print(y)
    continue

Submission status: Runtime Error

Solution code:
import math

n=int(input())
l=[int(input()) for i in range(n+1)]
def factor(m):
    num=m
    res=[]
    i=2
    while i*i<=m:
        if num%i:
            i+=1
        else:
            res.append(i)
            num//=i
    if num!=1:
        res.append(num)
    return res
res=list(set(factor(l[-1])))
#print(res)

def primes(n):
    is_prime = [True] * (n + 1)
    is_prime[0] = False
    is_prime[1] = False
    for i in range(2, int(n**0.5) + 1):
        if not is_prime[i]:
            continue
        for j in range(i * 2, n + 1, i):
            is_prime[j] = False
    return [i for i in range(n + 1) if is_prime[i]]
    
prime_lis=primes(10**6)
#print(prime_lis)
s=[]
for i in prime_lis:
    cnt=0
    for j in range(n+1):
        cnt+=l[n-j]*i**j
    s.append(cnt)
res.sort()
for i in res:
    check=True
    for j in s:
        if j%i!=0:
            check=False
            break
    if check:
        print(i)

Submission status: Runtime Error

Solution code:
def make_divisors(n):
    divisors = []
    for i in range(1, int(n**0.5)+1):
        if n % i == 0:
            divisors.append(i)
            if i != n // i:
                divisors.append(n//i)

    # divisors.sort()
    return divisors

def main():
    n=int(input())
    list=[]
    for i in range(n+1):
        list.append(int(input()))
    #print(list)
    count=1
    daihyo=list[-1]
    while list[-count]==0:
        count+=1
    yakusu=make_divisors(abs(list[-count]))
    #print(yakusu)
    ok=[]
    
    for yk in yakusu:
        amari=0
        amari2=0
        #print("yk:",yk)
        for l in range(len(list)):
            now=list[l]
            now2=list[l]*(2**(n-1))
            #print(now, n-l)
            amari=((amari+now)%yk)
            amari2=((amari2+now2)%yk)
            #print("a:", amari)
        if amari==0 and amari2==0:
            #print("OK:", yk)
            ok.append(yk)
    
    ok=sorted(ok, reverse=True)
    #print(yakusu)
    
    if ok[-1]==1:
        ok=ok[:-1]
    #print(ok)
    ok2=[]
    for x in range(len(ok)):
        swit=0
        for y in range(len(ok)-x-1):
            if ok[x]%ok[x+y+1]==0:
                swit=1
        if swit==0:
            ok2.append(ok[x])
    ok2=sorted(ok2)
    for x in ok2:
        print(x)
    
main()

Submission status: Runtime Error

Solution code:
N=int(input())
t=[int(input()) for i in range(N+1)]
a=[t[-i-1] for i in range(N+1)]

def f(x):
    v=a[0]
    y=x
    for i in range(N):
        v+=a[i]*y
        y*=x
    return v

def primedecomp(n):
    prime_dic={}
    for i in range(2,n):
        if i**2>n:
            break
        if n%i==0:
            prime_dic[i]=0
        while n%i==0:
            n//=i
            prime_dic[i]+=1

    if n>1:
        prime_dic[n]=1
    return prime_dic

def lcm(a,b):
    if a<b:
        a,b=b,a
    if a%b==0:
        return b
    else:
        return lcm(b,a%b)
def check(p):
    for i in range(1,p+1):
        if f(i)%p!=0:
            return False
    return True

ps = list(primedecomp(abs(a[0])).keys())
from collections import defaultdict
l=a[0]
for i in range(N):
    l=lcm(l,a[i+1])
ans=[]
for p in ps:
    if l%p==0:
        ans.append(p)
    elif check(p):
        ans.append(p)
for p in ans:
    print(p)

Submission status: Runtime Error

Solution code:
n = int(input())
a = []
for _ in range(n+1):
    a.append(int(input()))

max_val = a[-1]

def getCandidate(max_val):
    prime = [True for it in range(max_val+1)]

    prime[0] = False
    prime[1] = False

    cnt = 0

    candidate = []

    while True:
        # find next prime
        for _ in range(max_val + 1):
            if cnt>max_val:
                return candidate
#                 print('return')

            if prime[cnt]==False:
                cnt+=1
            else:
                break
#         print(cnt)

        if a[-1]%cnt == 0:
            candidate.append(cnt)

        j = 2
        while j*cnt <= max_val:
            prime[j*cnt]=False
            j += 1
        cnt+=1

def f(x, a):
    ret = 0
    for i in range(len(a)):
        p = len(a) - 1 -i
        ret += a[i]*x**p
    
    return ret

candi = getCandidate(a[-1])
for p in candi:
    FLAG = True
    for i in range(3):
        ans = f(i, a)
        if ans%p != 0:
            FLAG = False
    if FLAG:
        print(p)