Problem:
Score : 600 points

Problem Statement

Given is a positive integer N.

We will choose an integer K between 2 and N (inclusive), then we will repeat the operation below until N becomes less than K.

Operation: if K divides N, replace N with N/K; otherwise, replace N with N-K.

In how many choices of K will N become 1 in the end?

Constraints

2 \leq N \leq 10^{12}

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

Print the number of choices of K in which N becomes 1 in the end.

Sample Input 1

6

Sample Output 1

3

There are three choices of K in which N becomes 1 in the end: 2, 5, and 6.

In each of these choices, N will change as follows:

When K=2: 6 \to 3 \to 1

When K=5: 6 \to 1

When K=6: 6 \to 1

Sample Input 2

3141

Sample Output 2

13

Sample Input 3

314159265358

Sample Output 3

9

Solution in Python (3.4.3) (judge verdict: Time Limit Exceeded)

from copy import deepcopy
import numpy as np
def factor(N):
    ans = []
    for i in range(1,int(np.sqrt(N))+1):
        if N%i == 0:
            ans.append(i)
            if N//i != i:
                ans.append(i)
    return ans

def check(K,N):
    while 1:
        if N%K == 0:
            N = N//K
        else:
            break
    if N%K == 1:
        return True
    else:
        return False

N = int(input())
ans = len(factor(N-1)[1:])
for K in factor(N)[1:]:
    if check(K,N):
        ans += 1

print(ans)