Jzzhu has picked n apples from his big apple tree. All the apples are numbered from 1 to n. Now he wants to sell them to an apple store. 

Jzzhu will pack his apples into groups and then sell them. Each group must contain two apples, and the greatest common divisor of numbers of the apples in each group must be greater than 1. Of course, each apple can be part of at most one group.

Jzzhu wonders how to get the maximum possible number of groups. Can you help him?


-----Input-----

A single integer n (1 ≤ n ≤ 10^5), the number of the apples.


-----Output-----

The first line must contain a single integer m, representing the maximum number of groups he can get. Each of the next m lines must contain two integers — the numbers of apples in the current group.

If there are several optimal answers you can print any of them.


-----Examples-----
Input
6

Output
2
6 3
2 4

Input
9

Output
3
9 3
2 4
6 8

Input
2

Output
0

Difficulty: VERY_HARD
Tags: Number theory

Constructive algorithms

Starter code:

Solutions:
n = int(input())
primes = [x for x in range(2, int(n / 2) + 1) if all((x % y != 0 for y in range(2, int(x ** 0.5) + 1)))]
primes = primes[::-1]
used = {}
for i in range(2, n + 1):
	used[i] = False

def solve(n, primes, used):
	outputs = []
	doubles = []
	counter = 0
	if n == 1:
		return (0, 0)
	for i in primes:
		multiples = []
		for x in range(1, n // i + 1):
			if used[i * x] == False:
				multiples.append(i * x)
				used[i * x] = True
		if len(multiples) % 2 == 0:
			for j in range(0, int(len(multiples)), 2):
				outputs.append((multiples[j], multiples[j + 1]))
				counter += 1
		else:
			multiples.pop(1)
			used[i * 2] = False
			for j in range(0, int(len(multiples)), 2):
				outputs.append((multiples[j], multiples[j + 1]))
				counter += 1
	return (outputs, counter)
a = solve(n, primes, used)
print(a[1])
if a[0] != 0:
	for i in a[0]:
		print(*i)


apples = int(input())
if apples <= 3:
	print(0)
else:
	halfpr = int(apples / 2)

	def primes(n):
		isPrime = [True for i in range(n + 1)]
		isPrime[0] = isPrime[1] = False
		idx = 2
		while idx * idx <= n:
			if isPrime[idx]:
				for i in range(idx * 2, n, idx):
					isPrime[i] = False
			idx += 1
		return isPrime
	primeslist = primes(halfpr)
	totallist = [False] * (apples + 1)
	applepairs = []
	for prime in range(len(primeslist) - 1, 1, -1):
		if primeslist[prime]:
			numprimes = int(apples / prime)
			primesx = [int(i * prime) for i in range(1, numprimes + 1) if not totallist[i * prime]]
			if len(primesx) % 2 == 1:
				primesx.remove(2 * prime)
			for pr in primesx:
				applepairs.append(pr)
				totallist[pr] = True
	print(int(len(applepairs) / 2))
	for t in range(int(len(applepairs) / 2)):
		print(applepairs[2 * t], applepairs[2 * t + 1])


import math
apples = int(input())
if apples <= 3:
	print(0)
else:
	halfpr = int(apples / 2)

	def primes(n):
		primeslistsa = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317]
		yesnoprime = [False, False] + [True] * (n - 1)
		t = 0
		while primeslistsa[t + 1] <= int(math.sqrt(n)):
			t += 1
		for x in range(t + 1):
			for sa in range(2, int(n / primeslistsa[x] + 1)):
				yesnoprime[primeslistsa[x] * sa] = False
		return yesnoprime
	primeslist = primes(halfpr)
	totallist = [False] * (apples + 1)
	applepairs = []
	for prime in range(len(primeslist) - 1, 1, -1):
		if primeslist[prime]:
			numprimes = int(apples / prime)
			primesx = [int(i * prime) for i in range(1, numprimes + 1) if not totallist[i * prime]]
			if len(primesx) % 2 == 1:
				primesx.remove(2 * prime)
			for pr in primesx:
				applepairs.append(pr)
				totallist[pr] = True
	print(int(len(applepairs) / 2))
	for t in range(int(len(applepairs) / 2)):
		print(applepairs[2 * t], applepairs[2 * t + 1])


def find_primes(p):
	nums = list(range(3, p + 1))
	divisors = [2]
	for num in nums:
		check = 0
		res = True
		while divisors[check] <= num ** (1 / 2):
			if num % divisors[check] == 0:
				res = False
				break
			check += 1
		if res:
			divisors.append(num)
	divisors.reverse()
	return divisors
num = int(input())
nums = set(range(2, num + 1))
check = dict()
for i in range(1, num + 1):
	check[i] = False
primes = find_primes(num // 2)
results = []
for prime in primes:
	if prime > num // 2:
		break
	considering = []
	for x in range(1, num // prime + 1):
		if not check.get(x * prime):
			considering.append(x * prime)
			check[x * prime] = True
	if len(considering) % 2 != 0:
		considering.remove(prime * 2)
		check[prime * 2] = False
	adding = [(considering[i], considering[i + 1]) for i in range(0, len(considering), 2)]
	results += adding
nums = [x for x in nums if x % 2 == 0 and (not check.get(x))]
if len(nums) % 2 != 0:
	nums.pop()
results += [(nums[i], nums[i + 1]) for i in range(0, len(nums), 2)]
print(len(results))
for res in results:
	print(res[0], res[1])


from math import *

def prm(x):
	s = int(sqrt(x + 0.5))
	for i in range(2, s + 1):
		if not x % i:
			return False
	return True

def ap(a):
	while len(a) >= 2:
		print(a[-1], a[-2])
		a.pop()
		a.pop()
(n, cnt, ans, vis, ansl, ansr) = (int(input()), 0, 0, [False] * 100100, [], [])
prime = filter(prm, range(3, n // 2 + 1))
for x in prime:
	(tcnt, nlst) = (0, [])
	for i in range(1, n // x + 1):
		if not vis[i * x]:
			vis[i * x] = True
			nlst.append(i)
			tcnt += 1
	ans += tcnt >> 1
	cnt += tcnt & 1
	if tcnt & 1:
		nlst = nlst[0:1] + nlst[2:]
		ansr.append(x << 1)
	ansl += list(map(lambda k: x * k, nlst))
for i in range(1, n + 1):
	if not vis[i] and (not i & 1):
		ansr.append(i)
		cnt += 1
print(ans + (cnt >> 1))
ap(ansl)
ap(ansr)


def read(mode=2):
	inputs = input().strip()
	if mode == 0:
		return inputs
	if mode == 1:
		return inputs.split()
	if mode == 2:
		return [int(x) for x in inputs.split()]

def write(s='\n'):
	if isinstance(s, list):
		s = ' '.join(map(str, s))
	s = str(s)
	print(s, end='')
from itertools import compress
import itertools

def croft():
	for p in (2, 3, 5):
		yield p
	roots = {9: 3, 25: 5}
	primeroots = frozenset((1, 7, 11, 13, 17, 19, 23, 29))
	selectors = (1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0)
	for q in compress(itertools.islice(itertools.count(7), 0, None, 2), itertools.cycle(selectors)):
		if q in roots:
			p = roots[q]
			del roots[q]
			x = q + 2 * p
			while x in roots or x % 30 not in primeroots:
				x += 2 * p
			roots[x] = p
		else:
			roots[q * q] = q
			yield q
(n,) = read()
cr = croft()
primes = []
for i in cr:
	if i < n:
		primes.append(i)
	else:
		break
primes.reverse()
used = [0] * (n + 1)
res = []
for p in primes:
	k = n // p
	tmp = []
	while k:
		if not used[k * p]:
			tmp.append(k * p)
			used[k * p] = 1
		if len(tmp) == 2:
			res.append(tmp)
			tmp = []
		k -= 1
	if tmp == [p] and p > 2 and (p * 2 <= n) and len(res) and (res[-1][1] == p * 2):
		res[-1][1] = p
		used[p * 2] = 0
		used[p] = 1
print(len(res))
for i in res:
	print(' '.join(map(str, i)))


import math, sys, bisect, heapq
from collections import defaultdict, Counter, deque
from itertools import groupby, accumulate
input = iter(sys.stdin.buffer.read().decode().splitlines()).__next__
ilele = lambda : map(int, input().split())
alele = lambda : list(map(int, input().split()))

def list2d(a, b, c):
	return [[c] * b for i in range(a)]

def list3d(a, b, c, d):
	return [[[d] * c for j in range(b)] for i in range(a)]
MOD = 1000000000 + 7

def Y(c):
	print(['NO', 'YES'][c])

def y(c):
	print(['no', 'yes'][c])

def Yy(c):
	print(['No', 'Yes'][c])

def SieveOfEratosthenes(n):
	Z = []
	prime = [True for i in range(n + 1)]
	p = 2
	while p * p <= n:
		if prime[p] == True:
			for i in range(p * p, n + 1, p):
				prime[i] = False
		p += 1
	for p in range(2, n):
		if prime[p]:
			Z.append(p)
	return Z
N = int(input())
prime = SieveOfEratosthenes(N // 2)
if N <= 3:
	print(0)
else:
	mark = {}
	if N // 2 not in prime:
		prime.append(N // 2)
	Ans = []
	prime = prime[::-1]
	for i in prime:
		Z = []
		for j in range(1, N // i + 1):
			if mark.get(i * j, -1) == -1:
				Z.append(i * j)
		if len(Z) % 2 == 0:
			for k in Z:
				Ans.append(k)
				mark[k] = 1
		else:
			for k in Z:
				if k != 2 * i:
					Ans.append(k)
					mark[k] = 1
	print(len(Ans) // 2)
	for i in range(0, len(Ans), 2):
		print(Ans[i], Ans[i + 1])