Problem:


Score : 400 points

Problem Statement

You are given positive integers N and M.

How many sequences a of length N consisting of positive integers satisfy a_1 \times a_2 \times ... \times a_N = M? Find the count modulo 10^9+7.

Here, two sequences a' and a'' are considered different when there exists some i such that a_i' \neq a_i''.

Constraints

All values in input are integers.

1 \leq N \leq 10^5

1 \leq M \leq 10^9

Input

Input is given from Standard Input in the following format:

N M

Output

Print the number of the sequences consisting of positive integers that satisfy the condition, modulo 10^9 + 7.

Sample Input 1

2 6

Sample Output 1

4

Four sequences satisfy the condition: \{a_1, a_2\} = \{1, 6\}, \{2, 3\}, \{3, 2\} and \{6, 1\}.

Sample Input 2

3 12

Sample Output 2

18

Sample Input 3

100000 1000000000

Sample Output 3

957870001



Sample input:
2 6


Sample output:
4

Submission status: Wrong Answer

Solution code:
import math
MOD=10**9 + 7
N,M=map(int,input().split())
plist = 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dic = {}
sqm = math.sqrt(M)
sqm = M
plist = [p for p in plist if p <= sqm]
for p in plist:
    dic[p] = 0
for p in plist:
    while M%p == 0:
        M/=p
        dic[p]+=1
L = N + 40
fact = [1 for _ in range(L)]
invfact = [1 for _ in range(L)]
for i in range(1,L):
    fact[i]=(fact[i-1]*i)%MOD
    invfact[i]=(invfact[i-1]*pow(i,MOD-2,MOD))%MOD
ans = 1
for p in dic:
    if dic[p] !=0:
        ans *= fact[N-1+dic[p]]*invfact[N-1]*invfact[dic[p]] %MOD
print(ans%MOD)

Submission status: Wrong Answer

Solution code:
# encoding: utf-8
import math
N, M = map(int, input().split())

def distribute(pos, full, tmp):
    return memo[tmp][pos]

# define search range
p_max = M // 2 # M is prime if M has a factor p (> p_max)
ip_up = max((1, int(math.log2(M)))) # MDE

#### O(MlogM) below (if M has a very large prime factor)
M_tmp = M
ips = []
for p in range(2, p_max + 1):
    if M_tmp % p > 0: continue
    # how many times M_tmp can be divided
    p_tmp = p
    for i_p in range(ip_up + 1):
        if M_tmp % p_tmp > 0: break
        p_tmp = p_tmp * p
    # calculate later
    ips.append(i_p)
    # next
    M_tmp = M_tmp // (p_tmp // p)
    if M_tmp == 1: break
#### O(Mlog M) above

#### O(N) below
# create memo
## init
if len(ips) > 0: ip_max = max(ips)
else: ip_max = 0
memo = [[None] * N for i in range(ip_max + 1)] # max ip
for j in range(N): memo[0][j] = 1
for i in range(1, ip_max + 1): memo[i][N - 1] = 1
## fill
for i in range(1, ip_max + 1):
    sum_tmp = 0
    vals = list(memo[i - 1])[::-1]
    p_memo = memo[i]
    for j, val in enumerate(vals):
        sum_tmp += val
        p_memo[N - 1 - j] = sum_tmp
#### O(N) above
        
# calculate ans
ans = 1
for i_p in ips: ans = (ans * distribute(0, i_p, i_p)) % (10 ** 9 + 7)

if ip_max < 1: print(N)
else: print(ans)

Submission status: Wrong Answer

Solution code:
# encoding: utf-8
import math
N, M = map(int, input().split())

def distribute(pos, full, tmp):
    return memo[tmp][pos]

# define search range
p_max = int(M / 2) # M is prime if M has a factor p (> p_max)
ip_up = max((1, int(math.log2(M / 2)))) # MDE

#### O(MlogM) below (if M has a very large prime factor)
M_tmp = M
ips = []
for p in range(2, p_max + 1):
    if M_tmp % p > 0: continue
    # how many times M_tmp can be divided
    p_tmp = p
    for i_p in range(ip_up + 1):
        if M_tmp % p_tmp > 0: break
        p_tmp = p_tmp * p
    # calculate later
    ips.append(i_p)
    # next
    M_tmp = M_tmp // (p_tmp // p)
    if M_tmp == 1: break
#### O(Mlog M) above

#### O(N) below
# create memo
## init
if len(ips) > 0: ip_max = max(ips)
else: ip_max = 0
memo = [[None] * N for i in range(ip_max + 1)] # max ip
for j in range(N): memo[0][j] = 1
for i in range(1, ip_max + 1): memo[i][N - 1] = 1
## fill
for i in range(1, ip_max + 1):
    sum_tmp = 0
    vals = list(memo[i - 1])[::-1]
    p_memo = memo[i]
    for j, val in enumerate(vals):
        sum_tmp += val
        p_memo[N - 1 - j] = sum_tmp
#### O(N) above
        
# calculate ans
ans = 1
for i_p in ips: ans = (ans * distribute(0, i_p, i_p)) % (10 ** 9 + 7)

if ip_max < 1: print(N)
else: print(ans)

Submission status: Wrong Answer

Solution code:
# encoding: utf-8
import math
N, M = map(int, input().split())

def distribute(pos, full, tmp):
    return memo[tmp][pos]

# define search range
p_max = int(M / 2) # M is prime if M has a factor p (> p_max)
ip_max = int(math.log2(M))

M_tmp = M
ips = []
for p in range(2, p_max + 1):
    # how many times M_tmp can be divided
    p_tmp = p
    for i_p in range(ip_max + 1):
        if M_tmp % p_tmp > 0: break
        p_tmp = p_tmp * p
    # escape if M_tmp cannot be divided
    if i_p < 1: continue
    # calculate later
    ips.append(i_p)
    # next
    M_tmp = M_tmp // (p_tmp // p)
    if M_tmp == 1: break

### test code
print("WA")

Submission status: Wrong Answer

Solution code:
from collections import Counter

mod = 1000000007

# nの素因数分解

def prime(n):
    d = Counter()
    i = 2
    while i*i <= n:
        while n%i == 0:
            n //= i
            d[i] += 1
        i += 1
    if n>0:
        d[n] += 1
    return d

# xのn乗を計算する
def mod_pow(x, n):
    if n == 0:
        return 1
    elif n % 2 == 0:
        half_x = mod_pow(x, n // 2)
        return half_x * half_x % mod
    else:
        return x * mod_pow(x, n-1) % mod

fact = [1] * 100099
inv = [1] * 100099
for k in range(1, 100099):
    fact[k] = fact[k-1] * k % mod
    inv[k] = mod_pow(fact[k], mod-2)

def nCr(n, r):
    return fact[n] * inv[r] % mod * inv[n-r] % mod

def solve():
    N, M = map(int, input().split())
    p = prime(M)
    s = 1
    for i in p.values():
        s *= nCr(i + N - 1, N - 1)
        s %= mod
    print(s)

solve()

Submission status: Wrong Answer

Solution code:
import numpy as np
N,M=list(map(int,input().split()))
n=10**5
prime=[i for i in range(n)]
prime[1]=0
for i in range(2,n):
    tmp =i*i
    if prime[tmp]==0 or tmp > n:
        break
    while tmp<n:
        prime[tmp]=0
        tmp+=i

prime=np.array(prime)
prime=prime[prime!=0]

dic={}
for x in prime:
    while M%x==0:
        if dic.get(x,0)==0:
            dic[x]=1
        else:
            dic[x]+=1
        M/=x
    if M==1:
        break
ans = 1
for x in dic.values():
    tmp1=1
    tmp2=1
    for i in range(x,0,-1):
        tmp1*=(i+N-1)
        tmp2*=i
    ans*=tmp1//tmp2
    ans%=10**9+7
print(int(ans))

Submission status: Wrong Answer

Solution code:
N, M = map(int, input().split())

mod = 1000000007

f = [] # 素因数分解(指数部のみ)

for p in range(2, M): # 素因数分解
    e = 0
    while M%p == 0:
        e += 1
        M /= p
    
    if e>0:
        f.append(e)
    
    if M == 1:
        break

def pw(a, e):
    if e == 1:
        return a
    p = pw(a, e // 2)
    if e%2 == 1:
        return p * p * a % mod
    else:
        return p * p % mod

# fact, inv の計算
fact = [1] * 100040
inv = [1] * 100040
for k in range(1, 100040):
    fact[k] = fact[k-1] * k % mod
    inv[k] = pw(fact[k], mod-2)

ans = 1
for e in f:
    ans *= fact[e+N-1] * inv[e] * inv[N-1] % mod # N_H_e (重複組合せ)

print(ans)

Submission status: Wrong Answer

Solution code:
import numpy as np
N,M=list(map(int,input().split()))
n=10**5
prime=[i for i in range(n)]
prime[1]=0
for i in range(2,n):
    tmp =i*i
    if prime[tmp]==0 or tmp > n:
        break
    while tmp<n:
        prime[tmp]=0
        tmp+=i

prime=np.array(prime)
prime=prime[prime!=0]

dic={}
for x in prime:
    while M%x==0:
        if dic.get(x,0)==0:
            dic[x]=1
        else:
            dic[x]+=1
        M/=x
    if M==1:
        break
ans = 1
for x in dic.values():
    arr1=[]
    arr2=[]
    for i in range(x,0,-1):
        arr1.append(i+N-1)
        arr2.append(i)
    arr2.reverse()
    for i,y in enumerate(arr1):
        for j,z in enumerate(arr2):
            if arr1[i]%arr2[j]==0:
                arr1[i]/=arr2[j]
                arr2[j]=1.0
    for y in arr1:
        ans*=y
        ans%=10**9+7
print(int(ans))