Buggy python submission (verdict: Time Limit Exceeded):
#How many ways M=a1*a2*...*aN ex)N=2,M=6 a={1,6},{2,3},{3,2},{6,1}
import math
n,m=map(int,input().split())
factor=[]
c=0
mod=10**9+7

#Prime Factorization [number,times]
while m%2 == 0:
    m//=2
    c+=1
if c != 0:
    factor.append([2,c])
#Alternate
for i in range(3,m+1,2):
    c=0
    while m%i == 0:
        m//=i
        c+=1
    if c != 0:
        #print(i,c)
        factor.append([i,c])
    if m == 1:
        break
    if i*i>=m:  # m is prime
        factor.append([m,1])
        break
        
#print(factor)

def C(n,k):
    ans=1
    for s in range(1,k+1):
        ans*=n
        ans//=s
        n-=1
    return ans

def funcount(length,rest):
    a=0
    if length==1:
        #print(rest)
        return 1
    elif rest <= 1:
        #print("not",rest)
        return 0
    else:
        for i in range(1,rest):
            #print(i,end=" ")
            a+=funcount(length-1,rest-i)
        return a


way=1
for f in factor:
    sum=0
    for k in range(1,f[1]+1):
        sum+=funcount(k,f[1])*C(n,k)
    way=(way*sum)%mod
    
print(way%mod)

Fixed submission:
#How many ways M=a1*a2*...*aN ex)N=2,M=6 a={1,6},{2,3},{3,2},{6,1}
import math
n,m=map(int,input().split())
factor=[]
c=0
mod=10**9+7

#Prime Factorization [number,times]
while m%2 == 0:
    m//=2
    c+=1
    '''
    if c>=20:
        c=0
        break
        '''
if c != 0:
    factor.append([2,c])
#Alternate
for i in range(3,m+1,2):
    c=0
    while m%i == 0:
        m//=i
        c+=1
    if c != 0:
        #print(i,c)
        factor.append([i,c])
    if m == 1:
        break
    if i*i>=m:  # m is prime
        factor.append([m,1])
        break
        
#print(factor)

def C(n,k):
    ans=1
    for s in range(1,k+1):
        ans*=n
        ans//=s
        n-=1
    return ans

def funcount(length,rest,memo):
    a=0
    if length==1:
        #print(length,rest)
        memo[length][rest]=1
        return 1,memo
    elif rest <= 1:
        #print(length,rest)
        memo[length][rest]=0
        return 0,memo
    if memo[length][rest] != -1:
        return memo[length][rest],memo
    else:
        for i in range(1,rest):
            #print(i,end=" ")
            a+=funcount(length-1,rest-i,memo)[0]
            memo=funcount(length-1,rest-i,memo)[1]
            #print(memo)
        memo[length][rest]=a
        return a,memo
    

way=1
for f in factor:
    sum=0
    for k in range(1,f[1]+1):
        memo=[[-1 for r in range(f[1]+1)] for l in range(k+1)] 
        #print(memo)
        sum+=funcount(k,f[1],memo)[0]*C(n,k)
    way=(way*sum)%mod
    
print(way%mod)

Buggy python submission (verdict: Time Limit Exceeded):
#How many ways M=a1*a2*...*aN ex)N=2,M=6 a={1,6},{2,3},{3,2},{6,1}
import math
n,m=map(int,input().split())
factor=[]
c=0
mod=10**9+7

#Prime Factorization [number,times]
while m%2 == 0:
    m//=2
    c+=1
if c != 0:
    factor.append([2,c])
#Alternate
for i in range(3,m+1,2):
    c=0
    while m%i == 0:
        m//=i
        c+=1
    if c != 0:
        factor.append([i,c])


def C(n,k):
    ans=1
    for s in range(1,k+1):
        ans*=n
        ans//=s
        n-=1
    return ans

def funcount(length,rest):
    a=0
    if length==1:
        #print(rest)
        return 1
    elif rest <= 1:
        #print("not",rest)
        return 0
    else:
        for i in range(1,rest):
            #print(i,end=" ")
            a+=funcount(length-1,rest-i)
        return a


way=1
for f in factor:
    sum=0
    for k in range(1,f[1]+1):
        sum+=funcount(k,f[1])*C(n,k)
    way*=sum
    
print(way%mod)

Fixed submission:
#How many ways M=a1*a2*...*aN ex)N=2,M=6 a={1,6},{2,3},{3,2},{6,1}
import math
n,m=map(int,input().split())
factor=[]
c=0
mod=10**9+7

#Prime Factorization [number,times]
while m%2 == 0:
    m//=2
    c+=1
    '''
    if c>=20:
        c=0
        break
        '''
if c != 0:
    factor.append([2,c])
#Alternate
for i in range(3,m+1,2):
    c=0
    while m%i == 0:
        m//=i
        c+=1
    if c != 0:
        #print(i,c)
        factor.append([i,c])
    if m == 1:
        break
    if i*i>=m:  # m is prime
        factor.append([m,1])
        break
        
#print(factor)

def C(n,k):
    ans=1
    for s in range(1,k+1):
        ans*=n
        ans//=s
        n-=1
    return ans

def funcount(length,rest,memo):
    a=0
    if length==1:
        #print(length,rest)
        memo[length][rest]=1
        return 1,memo
    elif rest <= 1:
        #print(length,rest)
        memo[length][rest]=0
        return 0,memo
    if memo[length][rest] != -1:
        return memo[length][rest],memo
    else:
        for i in range(1,rest):
            #print(i,end=" ")
            a+=funcount(length-1,rest-i,memo)[0]
            memo=funcount(length-1,rest-i,memo)[1]
            #print(memo)
        memo[length][rest]=a
        return a,memo
    

way=1
for f in factor:
    sum=0
    for k in range(1,f[1]+1):
        memo=[[-1 for r in range(f[1]+1)] for l in range(k+1)] 
        #print(memo)
        sum+=funcount(k,f[1],memo)[0]*C(n,k)
    way=(way*sum)%mod
    
print(way%mod)