## Task

Given a positive integer, `n`, return the number of possible ways such that `k` positive integers multiply to `n`. Order matters.

**Examples**
```
n = 24
k = 2
(1, 24), (2, 12), (3, 8), (4, 6), (6, 4), (8, 3), (12, 2), (24, 1) -> 8

n = 100
k = 1
100 -> 1

n = 20
k = 3
(1, 1, 20), (1, 2, 10), (1, 4, 5), (1, 5, 4), (1, 10, 2), (1, 20, 1),
(2, 1, 10), (2, 2, 5), (2, 5, 2), (2, 10, 1), (4, 1, 5), (4, 5, 1),
(5, 1, 4), (5, 2, 2), (5, 4, 1), (10, 1, 2), (10, 2, 1), (20, 1, 1) -> 18
```
**Constraints**
`1 <= n <= 500_000_000`
and `1 <= k <= 1000`

Difficulty: MEDIUM_HARD
Tags: Matrices

Mathematics

Starter code:
def multiply(n, k):
	

Solutions:
from scipy.special import comb

def multiply(n, k):
	(r, d) = (1, 2)
	while d * d <= n:
		i = 0
		while n % d == 0:
			i += 1
			n //= d
		r *= comb(i + k - 1, k - 1, exact=True)
		d += 1
	if n > 1:
		r *= k
	return r


from scipy.special import comb
from collections import Counter

def factorize_exponents(n):
	result = Counter()
	while n % 2 == 0:
		result[2] += 1
		n //= 2
	k = 3
	m = int(n ** 0.5)
	while k <= m:
		if n % k == 0:
			result[k] += 1
			n //= k
			m = int(n ** 0.5)
		else:
			k += 2
	if n > 1:
		result[n] += 1
	return list(result.values())

def multiply(n, k):
	factorized = factorize_exponents(n)
	total = 1
	for exp in factorized:
		total *= comb(exp + k - 1, k - 1, exact=True)
	return total


from math import gcd, factorial
from random import randint

def primesfrom2to(n):
	sieve = [True] * (n // 2)
	for i in range(3, int(n ** 0.5) + 1, 2):
		if sieve[i // 2]:
			sieve[i * i // 2::i] = [False] * ((n - i * i - 1) // (2 * i) + 1)
	return [2] + [2 * i + 1 for i in range(1, n // 2) if sieve[i]]
bound = 7100
small_primes = primesfrom2to(bound)
small_primes_set = set(small_primes)

def factorint(n):
	factors = dict()
	limit = int(n ** 0.5) + 1
	for prime in small_primes:
		if prime > limit:
			break
		while n % prime == 0:
			if prime in factors:
				factors[prime] += 1
			else:
				factors[prime] = 1
			n //= prime
	if n >= 2:
		large_factors(n, factors)
	return factors

def large_factors(n, factors):
	while n > 1:
		if isprime(n):
			if n in factors:
				factors[n] += 1
			else:
				factors[n] = 1
			break
		factor = pollard_brent(n)
		large_factors(factor, factors)
		n //= factor

def pollard_brent(n):
	if n % 2 == 0:
		return 2
	(y, c, m) = (randint(1, n - 1), randint(1, n - 1), randint(1, n - 1))
	(g, r, q) = (1, 1, 1)
	while g == 1:
		x = y
		for i in range(r):
			y = (pow(y, 2, n) + c) % n
		k = 0
		while k < r and g == 1:
			ys = y
			for i in range(min(m, r - k)):
				y = (pow(y, 2, n) + c) % n
				q = q * abs(x - y) % n
			g = gcd(q, n)
			k += m
		r *= 2
	if g == n:
		while True:
			ys = (pow(ys, 2, n) + c) % n
			g = gcd(abs(x - ys), n)
			if g > 1:
				break
	return g

def isprime(n):
	if n % 2 == 0:
		return False
	elif n < bound:
		return n in small_primes_set
	(d, r) = (n - 1, 0)
	while d % 2 == 0:
		d //= 2
		r += 1
	for a in (2, 3, 5, 7):
		x = pow(a, d, n)
		if x == 1 or x == n - 1:
			continue
		for i in range(r - 1):
			x = pow(x, 2, n)
			if x == n - 1:
				break
		else:
			return False
	return True

def partitions(n, k):
	result = 1
	for i in range(n + 1, n + k):
		result *= i
	return result // factorial(k - 1)

def multiply(n, k):
	(total, factors) = (1, factorint(n))
	for exponent in list(factors.values()):
		total *= partitions(exponent, k)
	return total


def j(n, k):
	i = 1
	for f in range(1, k):
		i *= f + n
	for f in range(1, k):
		i //= f
	return i

def multiply(n, k):
	i = f = 1
	while f * f < n:
		(l, f) = (0, -~f)
		while n % f < 1:
			(l, n) = (-~l, n // f)
		if l:
			i *= j(l, k)
	return i * (n < 2 or j(1, k))


from collections import Counter
from functools import reduce
from operator import mul
small_primes = [2]

def is_prime(n):
	for p in small_primes:
		if p * p > n:
			return True
		if n % p == 0:
			return False
for n in range(3, 1000000):
	if is_prime(n):
		small_primes.append(n)

def factor(n):
	if n == 1:
		return Counter()
	for p in small_primes:
		if n % p != 0:
			continue
		factors = factor(n // p)
		factors[p] += 1
		return factors
	return Counter({n: 1})

def count_partitions(k, r):
	numerator = reduce(mul, list(range(r + 1, r + k)), 1)
	denominator = reduce(mul, list(range(1, k)), 1)
	return numerator // denominator

def multiply(n, k):
	factors = factor(n)
	print(n)
	print(factors)
	return reduce(mul, [count_partitions(k, r) for r in list(factors.values())], 1)


from math import sqrt, factorial

def c(n, m):
	return factorial(n) // (factorial(n - m) * factorial(m))

def factor(n):
	factors = {}
	max = int(sqrt(n) + 1)
	f = 2
	while f <= max:
		if n % f == 0:
			factors[f] = 0
			while n % f == 0:
				factors[f] += 1
				n //= f
			max = int(sqrt(n) + 1)
		f += 1
	if n != 1:
		factors[n] = 1
	return factors

def multiply(n, k):
	mul = 1
	for m in factor(n).values():
		mul *= c(m + k - 1, k - 1)
	return mul


import operator as op
from functools import reduce

def factors(n):
	div = 2
	d = dict()
	while n > (div - 1) ** 2:
		if n % div == 0:
			n = n // div
			if div in d:
				d[div] += 1
			else:
				d[div] = 1
		else:
			div += 1
	if n in d:
		d[n] += 1
	elif n > 1:
		d[n] = 1
	r = []
	for k in d.keys():
		r.append((k, d[k]))
	return r

def get_pascal(r):
	if r == 1:
		return [1]
	elif r == 2:
		return [1, 1]
	s = [1, 1]
	while len(s) < r:
		ns = [1]
		for i in range(len(s) - 1):
			ns.append(s[i] + s[i + 1])
		ns = ns + [1]
		s = ns
	return s

def ncr(n, r):
	if r > n:
		return 0
	r = min(r, n - r)
	numer = reduce(op.mul, range(n, n - r, -1), 1)
	denom = reduce(op.mul, range(1, r + 1), 1)
	return numer // denom

def get_cases(r, k):
	p = get_pascal(r)
	sum_v = 0
	for (i, v) in enumerate(p):
		sum_v += v * ncr(k, i + 1)
	return sum_v

def multiply(n, k):
	if n == 1:
		return 1
	f = factors(n)
	s = 1
	for r in f:
		s *= get_cases(r[1], k)
	return s


from collections import Counter
from functools import lru_cache, reduce
from itertools import combinations
from operator import mul
from math import factorial

def multiply(n, k):
	if k < 2:
		return k
	if k == 2:
		return len(factors(n))
	tups = get_factor_tuples(n, k)
	tups = {tuple((1,) * (k - len(x)) + x) for x in tups}
	f = factorial(k)
	return sum((f // reduce(mul, [factorial(v) for v in list(Counter(p).values()) if v > 1] + [1]) for p in tups))

def get_factor_tuples(n, k):
	pair = {(n,)}
	for kp in range(1, k):
		k_tup = set((x for x in pair if len(x) == kp))
		for grp in k_tup:
			for (i, e) in enumerate(grp):
				g = grp[:i] + grp[i + 1:]
				if e not in primefactors(n):
					for a in factors(e):
						if 1 < a < e:
							b = e // a
							pair.add(tuple(sorted((a, b) + g)))
	return pair

@lru_cache(maxsize=None)
def primefactors(n):
	(i, f) = (3, [])
	while n & 1 == 0:
		f.append(2)
		n = n >> 1
	while i * i <= n:
		if n % i:
			i += 2
		else:
			f.append(i)
			n //= i
	if n > 1:
		f.append(n)
	return f

@lru_cache(maxsize=None)
def factors(n):
	pf = primefactors(n)
	return {1, n} | {reduce(mul, x) for z in range(1, len(pf)) for x in combinations(pf, z)}