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: Accepted

Solution code:
def framod(n,mod):
    a=1
    for i in range(1,n+1):
        a=a * i % mod
    return a

def power(n,r,mod):
    if r == 0: return 1
    if r%2 == 0:
        return power(n*n % mod, r//2, mod) % mod
    if r%2 == 1:
        return n * power(n, r-1, mod) % mod
    
def comb(n,k,mod):
    if n<0 or k<0 or n<k: return 0
    if n==0 or k==0: return 1
    a=framod(n,mod)
    b=framod(k,mod)
    c=framod(n-k,mod)
    return (a * power(b, mod-2, mod) * power(c, mod-2, mod)) % mod

n,m=map(int, input().split())
pf={}
for i in range(2,int(m**0.5)+1):
    while m%i==0:
        pf[i]=pf.get(i,0)+1
        m//=i
if m>1:pf[m]=1

mod=10**9+7   
ans=1
for i in pf:
    ans=(ans * comb(pf[i]+n-1,pf[i],mod)) % mod
print(ans)

Submission status: Accepted

Solution code:
from collections import Counter
def primes(n):
  i = 2
  table = []
  while i * i <= n:
    while n % i == 0:
      n = n // i
      table.append(i)
    i += 1
  if n > 1:
    table.append(n)
  return table

def comb(m,n):
  if n > m - n:
    n = m - n
  r = 1
  for k in range(m, m - n, -1):
    r *= k
  d = 1
  for k in range(1, n + 1):
    d *= k
  return r // d

N,M = map(int, input().split())
ans = 1
for v in Counter(primes(M)).values():
  ans *= comb(N+v-1,N-1)
  ans %= 10**9+7
print(ans)

Submission status: Accepted

Solution code:
n,m=map(int,input().split())
pw = []
d = 2
while d*d <= m:
  e = 0
  while m % d == 0:
    e += 1
    m //= d
  if e > 0: pw.append(e)
  d += 1
if m > 1: pw.append(1)
ans = 1
for e in pw:
  comb = 1
  for i in range(e):
    comb = comb * (n+i) // (1+i)
  ans = ans * comb % 1000000007
print(ans)

Submission status: Accepted

Solution code:
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)

Submission status: Accepted

Solution code:
from math import floor, sqrt
from collections import Counter

MOD = 10 ** 9 + 7

# xを素因数分解する
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

# xのn乗（二分累乗法）
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[x]: xの階乗
facts = [1] + [0] * N2
for x in range(1, N2 + 1):
    facts[x] = (facts[x - 1] * x) % MOD

# invFs[x]: xの階乗の逆元
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)

Submission status: Accepted

Solution code:
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)

Submission status: Accepted

Solution code:
mod = 1000000007

# aのp乗を求めるアルゴリズム
# p=62のとき、pの変化は、62->31->30->15->14->7->6->3->2->1->0
# と、半分半分になっていく
# よって、計算量はO(logp)になる
def modpow(a, p):
    if p == 0:
        return 1
    if p % 2 == 0:
        # pが偶数のとき
        halfP = p // 2
        half = modpow(a, halfP)

        # a^(p/2) をhalfとして、half*halfを計算
        return half * half % mod
    else:
        # pが奇数のときは、
        # pを偶数にするために1減らす
        return a * modpow(a, p - 1) % mod


# (10*9*8) / (3*2*1)
# 10*9*8 -> ansMul
# 3*2*1 -> ansDiv
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
    # ansMul / ansDivをやりたい
    # ansDivの逆元を使って求める
    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
    # cntが2^Xとか3^XのXの部分
    ans *= calcComb(n + cnt - 1, n - 1) # N-1はcntと同じ
    ans %= mod
    i += 1
if MNkori != 1:
    # 最後に素数が残ってる分を処理する
    ans *= calcComb(n, n - 1)  # N-1はcntと同じ
    ans %= mod
print(ans)

Submission status: Accepted

Solution code:
from collections import Counter

mod = 1000000007

# nの素因数分解
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


# xのn乗を計算する
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()

Submission status: Accepted

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 > 1:
        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] * 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()