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

Solution (Python):
MOD=10**9 + 7
N,M=map(int,input().split())
plist = 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dic = {}
for p in plist:
    dic[p] = 0
for p in plist:
    while M%p == 0:
        M/=p
        dic[p]+=1
if M!=1:
    dic[M] = 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
        ans = ans%MOD
print(ans)

Solution (Python):
from math import floor, sqrt
from collections import Counter

MOD = 10 ** 9 + 7


def getPrimeFactor(x):
    ans = []
    for d in range(2, floor(sqrt(x)) + 1):
        while x % d == 0:
            ans.append(d)
            x //= d

    if x != 1:
        ans.append(x)

    return ans


def power(x, n):
    ans = 1
    while n:
        if n % 2 == 1:
            ans = (ans * x) % MOD
        x = (x * x) % MOD
        n //= 2
    return ans


N, M = map(int, input().split())

PFs = getPrimeFactor(M)
cnt = Counter(PFs)

N2 = N
if len(cnt):
    N2 += max(cnt.values())


facts = [1] + [0] * N2
for x in range(1, N2 + 1):
    facts[x] = (facts[x - 1] * x) % MOD


invFs = [0] * N2 + [power(facts[N2], MOD - 2)]
for x in reversed(range(N2)):
    invFs[x] = (invFs[x + 1] * (x + 1)) % MOD

def comb(n, k):
    return ((facts[n] * invFs[k]) % MOD * invFs[n - k]) % MOD


ans = 1
for num in cnt.values():
    ans = (ans * comb(N + num - 1, num)) % MOD
print(ans)

Solution (Python):
N, M = map(int, input().split())
 
def factorized(M):
    d = 2
    f = []
    while d <= M:
        count = 0
        while M % d == 0:
            M //= d
            count += 1
        if count > 0:
            f.append(count)
        d += 1
    return f
 
def tetra(n, m):
    x = 1
    for y in range(1, n):
        x = x * (m + y - 1) // y
    return x
 
x = 1
for f in factorized(M):
    x *= tetra(N, f + 1)
    x %= 10 ** 9 + 7
 
print(x)

Solution (Python):
mod = 1000000007





def modpow(a, p):
    if p == 0:
        return 1
    if p % 2 == 0:
        
        halfP = p // 2
        half = modpow(a, halfP)

        
        return half * half % mod
    else:
        
        
        return a * modpow(a, p - 1) % mod





def calcComb(a, b):
    if b > a - b:
        return calcComb(a, a - b)
    ansMul = 1
    ansDiv = 1
    for i in range(b):
        ansMul *= (a - i)
        ansDiv *= (i + 1)
        ansMul %= mod
        ansDiv %= mod
    
    
    ans = ansMul * modpow(ansDiv, mod - 2) % mod
    return ans


n, m = map(int, input().split())
MNkori = m
ans = 1
i = 2
while i * i <= MNkori:
    cnt = 0
    while MNkori % i == 0:
        cnt += 1
        MNkori //= i
    
    ans *= calcComb(n + cnt - 1, n - 1) 
    ans %= mod
    i += 1
if MNkori != 1:
    
    ans *= calcComb(n, n - 1)  
    ans %= mod
print(ans)

Solution (Python):
from collections import Counter

mod = 1000000007


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



def mod_pow(x, n):
    if n == 0:
        return 1
    elif n % 2 == 0:
        half = int(n / 2)
        half_x = mod_pow(x, half)
        return half_x * half_x % mod
    else:
        return x * mod_pow(x, n - 1) % mod


def nCr(n, r):
    x = 1
    r = min(r, n - r)
    for i in range(r):
        x *= n - i
        x %= mod
        x *= mod_pow(i + 1, mod - 2)
        x %= mod
    return x


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


solve()

Solution (Python):
from collections import Counter

mod = 1000000007



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


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] * 100100
inv = [1] * 100100
for k in range(1, 100100):
    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()

Solution (Python):
import numpy as np
N,M=list(map(int,input().split()))
n=4*10**4
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
if M !=1:
    dic[M]=1
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))

Solution (Python):
import math

n,m=map(int,input().split())
mod = 1000000007

if m==1:
    print(1)
    exit()

dic = {}
p = 2
while True:
    if m==1:
        break
    if p > math.floor(math.sqrt(m))+1:
        if dic.get(m) is None:
            dic[m] = 1
        else:
            dic[m] += 1
        break
    if m % p == 0:
        if dic.get(p) is None:
            dic[p] = 1
        else:
            dic[p] += 1
        m = m//p
        p=1
    p+=1

def pow(n,e):
    ret=1
    b=n
    while e > 0:
        if e % 2 == 1:
            ret = ret * b % mod
        b = b * b % mod
        e = e//2
    return ret
    
def inv(n):
    return pow(n,mod-2)

def nck(n,k):
    ret = 1
    for i in range(1,k+1):
        ret *= (n-i+1) % mod
        ret %= mod
        ret *= inv(i) % mod
        ret %= mod
    return ret
    
ret=1
for x in dic.keys():
    ret = (ret * nck(n+dic[x]-1,min(n-1,dic[x]))) % mod

print(ret)