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This solution with python , but has some test faild because of Time limit exceeded .

import math

import os

import random

import re

import sys

def power(x, y, p):

result = 1

x = x % p

while y > 0:

if y & 1:

result = (result * x) % p

y = y >> 1

x = (x * x) % p

return result

def is_primitive_root(g, p, factors):

for factor in factors:

if power(g, (p - 1) // factor, p) == 1:

return False

return True

def find_primitive_roots(p):

primitive_roots = []

factors = set()

# Find prime factors of p-1

for i in range(2, int(math.sqrt(p-1)) + 1):

if (p - 1) % i == 0:

factors.add(i)

factors.add((p - 1) // i)

for g in range(2, p):

if is_primitive_root(g, p, factors):

primitive_roots.append(g)

return primitive_roots

def smallest_primitive_root(p):

primitive_roots = find_primitive_roots(p)

return min(primitive_roots)

if name == 'main':

p = int(input().strip())

smallest_root = smallest_primitive_root(p)

total_roots = len(find_primitive_roots(p))

print(smallest_root, total_roots)

This solution with python , but has some test faild because of Time limit exceeded .

import math import os import random import re import sys

def power(x, y, p): result = 1 x = x % p while y > 0: if y & 1: result = (result * x) % p y = y >> 1 x = (x * x) % p return result

def is_primitive_root(g, p, factors): for factor in factors: if power(g, (p - 1) // factor, p) == 1: return False return True

def find_primitive_roots(p): primitive_roots = [] factors = set() # Find prime factors of p-1 for i in range(2, int(math.sqrt(p-1)) + 1): if (p - 1) % i == 0: factors.add(i) factors.add((p - 1) // i)

for g in range(2, p):
    if is_primitive_root(g, p, factors):
        primitive_roots.append(g)
return primitive_roots

def smallest_primitive_root(p): primitive_roots = find_primitive_roots(p) return min(primitive_roots)

if name == 'main': p = int(input().strip())

smallest_root = smallest_primitive_root(p)
total_roots = len(find_primitive_roots(p))

print(smallest_root, total_roots)

To find the smallest primitive root and the total number of primitive roots for a given prime number, we can use mathematical concepts. Primitive roots are fascinating mathematical objects, and they play a crucial role in number theory

Absolutely, mathematics is a powerful tool that I often use for various purposes. It helps me understand probabilities and make informed decisions in various situations. Learning mathematics has been essential for me, as it has enabled me to develop critical thinking and problem-solving skills that I can apply in my everyday life and in various fields of study.