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

Solution (Python):
def gcd(x,y):
    if x<y:
        x,y=y,x
    if y==0:
        return x
    if x%y==0:
        return y
    else:
        return gcd(x%y,y)
N=int(input())
a=[int(input()) for i in range(N+1)][::-1]


g=abs(a[0])
for i in range(1,N+1):
    g=gcd(g,abs(a[i]))
def check(P):
    b=[a[i]%P for i in range(1,N+1)]
    if a[0]%P!=0:
        return False
    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=set()
X=[1 for i in range(max(N+1,3))]
X[0]=0;X[1]=0
i=2
while(i*i<=N):
    for j in range(i,N+1,i):
        if i==j:
            if X[i]==0:
                break
        else:
            X[j]=0
    i+=1
i=2
tmp=g
while(i*i<=g):
    if tmp%i==0:
        Plist.add(i)
        while(1):
            if tmp%i==0:
                tmp=tmp//i
            else:
                break
    i+=1
if tmp>1:
    Plist.add(tmp)
for j in range(N+1):
    if X[j]==1:
        Plist.add(j)
Plist=sorted(list(Plist))
for p in Plist:
    if check(p):
        print(p)

Solution (Python):
def gcd(a, b):
	while a > 0 and b > 0:
		a, b = b, a % b
	return a + b

def st(a, b):
	if b == 0:
		return 1
	if b % 2 == 0:
		q = st(a, b // 2)
		return q * q
	return a * st(a, b - 1)

n = int(input())
a = []
for i in range(n + 1):
	x = int(input())
	a.append(x)
ans = 0
massiv = [1, 2, 3, 5, 7, 11, 13, 17, 19, 21, 23, 9, 16]
for x in massiv:
	tmp = 0
	r = st(x, n)
	for i in range(0, n + 1):
		tmp += a[i] * r
		r //= x
	ans = gcd(ans, abs(tmp))
v = []
j = 2
while j * j <= ans:
	if ans % j == 0:
		ans //= j
		v.append(j)
	else:
		j += 1
if ans != 1:
	v.append(ans)
v.sort()
for i in range(len(v)):
	if i == 0 or v[i] != v[i - 1]:
		print(v[i])

Solution (Python):
from functools import reduce
import random
MOD = 59490679579998635868564022242503445659680322440679327938309703916140405638383434744833220555452130558140772894046169700747186409941317550665821619479140098743288335106272856533978719769677794737287419461899922456614256826709727133515885581878614846298251428096512495590545989536239739285290509870860732435109794045491640561011775744617218426011793283330497878491137999673760740717904979730106994180691956740996319139767941335979146924614249205862051182094081815447356600355657858936442589201739350474320966093143098768914226528498860747050474426418211313776073365498029087492308102876323562641651940555529817223923967479711293692697592800750806084802623966596666156361791014935954720420383555300213829134029560826306201326905674141090667002959274513478562033467485802590354539887133866434256562999666615074220857502563011098759348850149774083443643246907501430076141377431759984926248272553982527014011245310690986644461251576837779050124866641531488351016070300247918750613207952783437248236577750579092103591540592540838395240861443615842939673889972504776455114767522430347610500271801944701491874649306605557851914677316523034726044356368337787113190245601235705959816327016988315321850324065292964680148884818699916224411245926491625245571963741062587277736372090007125908660590314049976206281064703285728879581199596401313352724403492043120507597057004633720111008838673185885941472392623805512709896864520875740497617898566981218781313900004406341154884180472153171064617953661517880143988867189622824081729539392205701820144922307223627345876707465251206005262622236311161449785941834002795854996108322186230425179217618301526151712928790928933798369678844576216735378555233905935973195721247604933753363412045618703247367192610615234835041956551569232557323060407648325565048712478527583315981204846341857095134567470182330491262285172727037299211391244340592936174221176781260586188162350081192408213470101717320475998863222409777310443027421196981193126541663212124245716187453863438039402316877152286468198891603632606578778749292403571792687832081974134637014026451921536576338243322267400651083805535119025415817887075652758045539565968044552126338330231434466204888993650859153585380124240540573308417330478048240203241631072371322849430883727355239704116556046700749530006852187064160849175332758172150251213637470549781080491037088372092203085237973008861896576796238915011886636658033019385943299986285181723378096425117379056207797455963451889971988904466449319007760192467211209692128894691704353648198130409333996534250878389064152054828983825841234644875996912485916827004219887033833599723481903489316488764021700996686817244736947119285629049355809027206179193628292018441744552168286541735687924729198455883895791600170372255284216915442808139396541702893732917062958054499525549626455191658842064247047426187897146172971001949767308335268505414284088528125611263685734457560292833389995980698745893243832547007166243476192958601735336260255598581701267151224204461879782815468518040925292817115377696676461775120750971210951527384637825092221708015393564320979357698186262039029460777050248162599194429941464248920952161182024344007059684970270762248243899640259750891406957836740989312390963091260380990901672119736141666856448171380781556117025832312710039041398538035351795267732240730608951176127282191528457681241590895740457571038936173983449289126574141189374690274057472401359482497502067814596008557725079835212621242944853319496441084441343866446380876967613370281793088540430288658573281302876742323336651987699532240686371751448290351615451932054274752105688318766958111191822471878078268490672607804265064578569581247796205593441336042254502454646980009177290888726099355974892680373341371214123450598691915802122913613669446370194221846225597401405625536693874723700374068542217340621938985167525416725638580266200550048001242788847319217369432802469735271132107428204697172144851088692772696511622350096452418244968500432645305761138012204888798724453956720733374364721511323104353496583927094760495687785031050687852300161433757934364774351673531859394855389527851678630166084819717354779333605871648837631164630550613112327670353108353791304451834904017538583880634608113426156225757314283269948216017294095002859515842577481167417167637534527111130468710

def gcd(a, b):
    b = abs(b)
    while b != 0:
        r = a%b
        a,b = b,r
    return a
def gcd_mult(numbers):
    return reduce(gcd, numbers)

N = int(input())
A = [int(input()) for _ in range(N+1)][::-1]
g = gcd_mult(A)

A = [a//g for a in A]

def f(n):
    ret = A[N]
    for i in range(N-1, -1, -1):
        ret = (ret * n + A[i]) % MOD
    return ret

ans = MOD
for _ in range(20):
    k = random.randrange(10**10)
    ans = gcd(ans, f(k))

def primeFactor(N):
    i = 2
    ret = {}
    n = N
    if n < 0:
        ret[-1] = 1
        n = -n
    if n == 0:
        ret[0] = 1
    d = 2
    sq = int(n ** (1/2))
    while i <= sq:
        k = 0
        while n % i == 0:
            n //= i
            k += 1
            ret[i] = k
        if k > 0:
            sq = int(n**(1/2))
        if i == 2:
            i = 3
        elif i == 3:
            i = 5
        elif d == 2:
            i += 2
            d = 4
        else:
            i += 4
            d = 2
        
    if n > 1:
        ret[n] = 1
    return ret
ANS = []
for i in primeFactor(ans*g):
    ANS.append(i)

ANS = sorted(ANS)
for ans in ANS:
    print(ans)

Solution (Python):
def gcd(a, b):
	while a > 0 and b > 0:
		a, b = b, a % b
	return a + b

def st(a, b):
	if b == 0:
		return 1
	if b % 2 == 0:
		q = st(a, b // 2)
		return q * q
	return a * st(a, b - 1)

n = int(input())
a = []
for i in range(n + 1):
	x = int(input())
	a.append(x)
ans = 0
for x in range(1, 18):
	tmp = 0
	r = st(x, n)
	for i in range(0, n + 1):
		tmp += a[i] * r
		r //= x
	ans = gcd(ans, abs(tmp))
v = []
j = 2
while j * j <= ans:
	if ans % j == 0:
		ans //= j
		v.append(j)
	else:
		j += 1
if ans != 1:
	v.append(ans)
v.sort()
for i in range(len(v)):
	if i == 0 or v[i] != v[i - 1]:
		print(v[i])

Solution (Python):
from functools import reduce
import random
import time
MOD = 59490679579998635868564022242503445659680322440679327938309703916140405638383434744833220555452130558140772894046169700747186409941317550665821619479140098743288335106272856533978719769677794737287419461899922456614256826709727133515885581878614846298251428096512495590545989536239739285290509870860732435109794045491640561011775744617218426011793283330497878491137999673760740717904979730106994180691956740996319139767941335979146924614249205862051182094081815447356600355657858936442589201739350474320966093143098768914226528498860747050474426418211313776073365498029087492308102876323562641651940555529817223923967479711293692697592800750806084802623966596666156361791014935954720420383555300213829134029560826306201326905674141090667002959274513478562033467485802590354539887133866434256562999666615074220857502563011098759348850149774083443643246907501430076141377431759984926248272553982527014011245310690986644461251576837779050124866641531488351016070300247918750613207952783437248236577750579092103591540592540838395240861443615842939673889972504776455114767522430347610500271801944701491874649306605557851914677316523034726044356368337787113190245601235705959816327016988315321850324065292964680148884818699916224411245926491625245571963741062587277736372090007125908660590314049976206281064703285728879581199596401313352724403492043120507597057004633720111008838673185885941472392623805512709896864520875740497617898566981218781313900004406341154884180472153171064617953661517880143988867189622824081729539392205701820144922307223627345876707465251206005262622236311161449785941834002795854996108322186230425179217618301526151712928790928933798369678844576216735378555233905935973195721247604933753363412045618703247367192610615234835041956551569232557323060407648325565048712478527583315981204846341857095134567470182330491262285172727037299211391244340592936174221176781260586188162350081192408213470101717320475998863222409777310443027421196981193126541663212124245716187453863438039402316877152286468198891603632606578778749292403571792687832081974134637014026451921536576338243322267400651083805535119025415817887075652758045539565968044552126338330231434466204888993650859153585380124240540573308417330478048240203241631072371322849430883727355239704116556046700749530006852187064160849175332758172150251213637470549781080491037088372092203085237973008861896576796238915011886636658033019385943299986285181723378096425117379056207797455963451889971988904466449319007760192467211209692128894691704353648198130409333996534250878389064152054828983825841234644875996912485916827004219887033833599723481903489316488764021700996686817244736947119285629049355809027206179193628292018441744552168286541735687924729198455883895791600170372255284216915442808139396541702893732917062958054499525549626455191658842064247047426187897146172971001949767308335268505414284088528125611263685734457560292833389995980698745893243832547007166243476192958601735336260255598581701267151224204461879782815468518040925292817115377696676461775120750971210951527384637825092221708015393564320979357698186262039029460777050248162599194429941464248920952161182024344007059684970270762248243899640259750891406957836740989312390963091260380990901672119736141666856448171380781556117025832312710039041398538035351795267732240730608951176127282191528457681241590895740457571038936173983449289126574141189374690274057472401359482497502067814596008557725079835212621242944853319496441084441343866446380876967613370281793088540430288658573281302876742323336651987699532240686371751448290351615451932054274752105688318766958111191822471878078268490672607804265064578569581247796205593441336042254502454646980009177290888726099355974892680373341371214123450598691915802122913613669446370194221846225597401405625536693874723700374068542217340621938985167525416725638580266200550048001242788847319217369432802469735271132107428204697172144851088692772696511622350096452418244968500432645305761138012204888798724453956720733374364721511323104353496583927094760495687785031050687852300161433757934364774351673531859394855389527851678630166084819717354779333605871648837631164630550613112327670353108353791304451834904017538583880634608113426156225757314283269948216017294095002859515842577481167417167637534527111130468710
sTime = time.time()

def gcd(a, b):
    b = abs(b)
    while b != 0:
        r = a%b
        a,b = b,r
    return a
def gcd_mult(numbers):
    return reduce(gcd, numbers)

N = int(input())
A = [int(input()) for _ in range(N+1)][::-1]
g = gcd_mult(A)

A = [a//g for a in A]

def f(n):
    ret = A[N]
    for i in range(N-1, -1, -1):
        ret = (ret * n + A[i]) % MOD
    return ret

ans = MOD
while time.time()-sTime < 1.5:
    k = random.randrange(10**10)
    ans = gcd(ans, f(k))

def primeFactor(N):
    i = 2
    ret = {}
    n = N
    if n < 0:
        ret[-1] = 1
        n = -n
    if n == 0:
        ret[0] = 1
    d = 2
    sq = int(n ** (1/2))
    while i <= sq:
        k = 0
        while n % i == 0:
            n //= i
            k += 1
            ret[i] = k
        if k > 0:
            sq = int(n**(1/2))
        if i == 2:
            i = 3
        elif i == 3:
            i = 5
        elif d == 2:
            i += 2
            d = 4
        else:
            i += 4
            d = 2
        
    if n > 1:
        ret[n] = 1
    return ret
ANS = []
for i in primeFactor(ans*g):
    ANS.append(i)

ANS = sorted(ANS)
for ans in ANS:
    print(ans)

Solution (Python):


import sys, collections, bisect, math, itertools
from fractions import gcd 


m = int(input())


a = [int(input()) for j in range(m+1)] 

beki = 0
while a[-1] == 0:
    a.pop()
    beki+=1
if m != len(a)-1:
    m = len(a)
    a.append(0)


def get_sieve_of_eratosthenes(n):
    if not isinstance(n, int):
        raise TypeError('n is int type.')
    if n < 2:
        raise ValueError('n is more than 2')
    prime = []
    limit = math.sqrt(n)
    data = [i + 1 for i in range(1, n)]
    while True:
        p = data[0]
        if limit <= p:
            return prime + data
        prime.append(p)
        data = [e for e in data if e % p != 0]
allprimes = get_sieve_of_eratosthenes(10010)

primes=[]
def get_sieve_of_eratosthenes2():
    prime = []
    data = [i + 1 for i in range(1, 35000)]
    c=abs(a[m])
    while True:
        p = data[0]
        if 10**9 <= p*p:
            return prime + data
        if c%p==0: primes.append(p)
        prime.append(p)
        data = [e for e in data if e % p != 0]

primes2=[]
def get_sieve_of_eratosthenes3():
    prime = []
    data = [i + 1 for i in range(1, 35000)]
    c=abs(a[0])
    while True:
        p = data[0]
        if 10**9 <= p*p:
            return prime + data
        if c%p==0: primes2.append(p)
        prime.append(p)
        data = [e for e in data if e % p != 0]


get_sieve_of_eratosthenes2()
get_sieve_of_eratosthenes3()
if len(primes) == 0 and abs(a[m]) != 1:
    primes.append(abs(a[m]))
if a[m]==0:
    primes = allprimes
if len(primes2) == 0 and abs(a[0]) != 1 and abs(a[0]) not in primes:
    primes2.append(abs(a[0]))
n=m


primes = sorted(list(set(primes + primes2)))


for p in primes:
    if p > 10**4+5:
        if sum([i%p for i in a]) == 0:
            print(p)
        continue
    coeff = list(reversed(a))
    while len(coeff) > p:

        t=coeff.pop()
        coeff[-p+1]+=t
            

    if sum([i%p for i in coeff]) == 0:
        print(p)