Jzzhu and Apples (Codeforces 450E)
Tags: Others, number theory

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 ≤ 105), 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.


Sample Inputs:
6

9

2

Sample Outputs:
2
6 3
2 4

3
9 3
2 4
6 8

0

Solution (Python):
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])

Solution (Python):
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])

Solution (Python):
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])