Problem:
 Score : 500 points    Problem Statement Let us denote by f(x, m) the remainder of the Euclidean division of x by m .  Let A be the sequence that is defined by the initial value A_1=X and the recurrence relation A_{n+1} = f(A_n^2, M) .
Find \displaystyle{\sum_{i=1}^N A_i} .      Constraints  1 \leq N \leq 10^{10}  0 \leq X < M \leq 10^5  All values in input are integers.         Input Input is given from Standard Input in the following format:  N  X  M       Output Print \displaystyle{\sum_{i=1}^N A_i} .        Sample Input 1 6 2 1001      Sample Output 1 1369  The sequence A begins 2,4,16,256,471,620,\ldots Therefore, the answer is 2+4+16+256+471+620=1369 .       Sample Input 2 1000 2 16      Sample Output 2 6  The sequence A begins 2,4,0,0,\ldots Therefore, the answer is 6 .       Sample Input 3 10000000000 10 99959      Sample Output 3 492443256176507

Solution (Python):
def main(sample_file = ''):

    """ convenient functions
    
    
    
    
    
    """

    """convenient decorator
    
    
        
        
        
        
        
        
        
        
        
        
    """
    
    
    import sys
    sys.setrecursionlimit(10**7)
    from itertools import accumulate, combinations, permutations, product, combinations_with_replacement 
    
    from math import factorial, ceil, floor, sqrt
    def factorize(n):
        """return the factors of the Arg and count of each factor
        
        Args:
            n (long): number to be resolved into factors
        
        Returns:
            list of tuples: factorize(220) returns [(2, 2), (5, 1), (11, 1)]
        """
        fct = []  
        b, e = 2, 0  
        while b * b <= n:
            while n % b == 0:
                n = n // b
                e = e + 1
            if e > 0:
                fct.append((b, e))
            b, e = b + 1, 0
        if n > 1:
            fct.append((n, 1))
        return fct
    def combinations_count(n, r):   
        """Return the number of selecting r pieces of items from n kinds of items.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        
        if n < 0 or r < 0:
            raise Exception('combinations_count(n, r) not defined when n or r is negative')
        if n - r < r: r = n - r
        if r < 0: return 0
        if r == 0: return 1
        if r == 1: return n
        numerator = [n - r + k + 1 for k in range(r)]
        denominator = [k + 1 for k in range(r)]
        for p in range(2,r+1):
            pivot = denominator[p - 1]
            if pivot > 1:
                offset = (n - r) % p
                for k in range(p-1,r,p):
                    numerator[k - offset] /= pivot
                    denominator[k] /= pivot
        result = 1
        for k in range(r):
            if numerator[k] > 1:
                result *= int(numerator[k])
        return result
    def combinations_with_replacement_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items allowing individual elements to be repeated more than once.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        if n < 0 or r < 0:
            raise Exception('combinations_with_replacement_count(n, r) not defined when n or r is negative')
        elif n == 0:
            return 1
        else:
            return combinations_count(n + r - 1, r)
    from bisect import bisect_left, bisect_right
    from collections import deque, Counter, defaultdict 
    from heapq import heapify, heappop, heappush, heappushpop, heapreplace,nlargest,nsmallest 
    from copy import deepcopy, copy 
    import operator
    from operator import itemgetter 
    
    
    from functools import reduce, lru_cache
    def chmin(x, y):
        """change minimum
        if x > y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current minimum value
            y (long): potential minimum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x > y:
            x = y
            return (x, True)
        else:
            return (x, False)
    def chmax(x, y):
        """change maximum
        if x < y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current maximum value
            y (long): potential maximum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x < y:
            x = y
            return (x, True)
        else:
            return (x, False)

    from fractions import gcd
    def gcd(a,b):
        
        
        if a<b:
            a, b = b, a
        while a%b !=0 :
            a, b = b, a%b
        return b

    
    
    
    
    
    

    def gcds(numbers):
        return reduce(gcd, numbers)
    def lcm(x, y):
        return (x * y) // gcd(x, y)
    def lcms(numbers):
        return reduce(lcm, numbers, 1)
    def make_divisors(n, reversed=False):
        """create list of divisors
        
        Args:
            number (int): number from which list of divisors is created
            reversed (bool, optional): ascending order if False. Defaults to False.
        
        Returns:
            list: list of divisors
        """
        divisors = set()
        for i in range(1, int(n**0.5)+1):
            if n % i == 0:
                divisors.add(i)
                divisors.add(n//i)
        return sorted(list(divisors),reverse=reversed)

    
    def sieve_of_eratosthenes(end, typecode="L"):
        
        

        
        import array
        import textwrap
        assert end > 1
        
        
        
        is_prime = array.array("B", (True for i in range(end)))
        
        is_prime[0] = False
        is_prime[1] = False
        
        primes = array.array(typecode)
        
        for i in range(2, end):
            if is_prime[i]:
                primes.append(i)
                for j in range(2 * i, end, i):
                    is_prime[j] = False  
        return primes

    class PrimeFactorization:
        '''初期化にO(NloglogN)/ 因数分解クエリをO(log(k))/ 素数判定クエリをO(1)
        >>> pf = PrimeFactorization(100)
        >>> pf.query(100)
        ((2, 2), (5, 2))
        
        '''
        
        
        
        
        
        
        
        def __init__(self, Num=10**6):
            from collections import defaultdict
            from math import floor
            self.Num = Num
            self.D = [-1] * (Num+1)
            for i in range(2, Num+1):
                if self.D[i] != -1:
                    continue
                self.D[i] = i
                for j in range(1, floor(Num/i)+1):
                    if self.D[j*i] != -1:
                        continue
                    else:
                        self.D[j*i] = i
        
        def query(self, k):
            ans = defaultdict(int)
            tmp = k
            while self.D[tmp] != -1:
                div = self.D[tmp]
                ans[div] += 1
                tmp = tmp//div
            return tuple(ans.items())
        
        def is_prime(self, k):
            return k == self.D[k]

    
    
    INF = 10 ** 18
    MOD = 10 ** 9 + 7
    AtoZ = list('abcdefghijklmnopqrsruvwxyz') 
    modpow = lambda a, n, p = MOD: pow(a, n, p) 
    def modinv(a, p = MOD):
        
        
        return modpow(a, p-2, p)
    def modinv_list(n, p = MOD):
        if n <= 1:
            return [0,1][:n+1]
        else:
            inv_t = [0,1]
            for i in range(2, n+1):
                inv_t += [inv_t[p % i] * (p - int(p / i)) % p]
            return inv_t
    def modfactorial_list(n, p = MOD):
        if n == 0:
            return [1]
        else:
            l = [0] * (n+1)
            tmp = 1
            for i in range(1, n+1):
                tmp = tmp * i % p
                l[i] = tmp
            return l
    def modcomb(n, k, fac_list = [], p = MOD):
        
        
        from math import factorial
        if n < 0 or k < 0 or n < k: return 0
        if n == 0 or k == 0: return 1
        if len(fac_list) <= n:
            a = factorial(n) % p
            b = factorial(k) % p
            c = factorial(n-k) % p
        else:
            a = fac_list[n]
            b = fac_list[k]
            c = fac_list[n-k]
        return (a * modpow(b, p-2, p) * modpow(c, p-2, p)) % p
    def modadd(a, b, p = MOD):
        return (a + b) % MOD
    def modsub(a, b, p = MOD):
        return (a - b) % p
    def modmul(a, b, p = MOD):
        return ((a % p) * (b % p)) % p
    def moddiv(a, b, p = MOD):
        return modmul(a, modpow(b, p-2, p))

    class UnionFindTree:
        """union find tree class
        TODO: fix this description...
        how to use (example):
        >>  uf = UnionFindTree(N) 
        >>  if uf.find_root(a) == uf.find_root(b):
        >>      do something like:
        >>      pass
        >>  else:
        >>      do something like:
        >>      L[find_root(a)] = L[find_root(a)] + L[find_root(b)]
        >>      L[find_root(b)] = L[find_root(a)]
        >>      
        >>      uf.unite(a, b)
        """
        def __init__(self, N):
            self.root = [-1] * (N+1)
            self.rank   = [0]  * (N+1)
            self.connected_num = [1] * (N+1)
    
        def find_root(self,x):
            root = self.root
            while root[x] != -1:
                x = root[x]
            return x
    
        def unite(self,x,y):
            root = self.root
            rank = self.rank
            connected_num = self.connected_num
            find_root = self.find_root
    
            rx = find_root(x)
            ry = find_root(y)
            if rx != ry:
                if rank[rx] < rank[ry]:
                    root[rx] = ry
                    rx,ry = ry,rx
                else:
                    if rank[rx] == rank[ry]:
                        rank[rx] += 1
                    root[ry] = rx       
                connected_num[rx] += connected_num[ry]

    
    
    
    class SegTree:
        """
        ref: https://qiita.com/takayg1/items/c811bd07c21923d7ec69
        init(init_val, ide_ele): 配列init_valで初期化 O(N)
        update(k, x): k番目の値をxに更新 O(logN)
        query(l, r): 区間[l, r)をsegfuncしたものを返す O(logN)
        """
        def __init__(self, init_val, segfunc, ide_ele):
            """
            init_val: 配列の初期値
            segfunc: 区間にしたい操作
            ide_ele: 単位元
            n: 要素数
            num: n以上の最小の2のべき乗
            tree: セグメント木(1-index)
            """
            n = len(init_val)
            self.segfunc = segfunc
            self.ide_ele = ide_ele
            self.num = 1 << (n - 1).bit_length()
            self.tree = [ide_ele] * 2 * self.num
            
            for i in range(n):
                self.tree[self.num + i] = init_val[i]
            
            for i in range(self.num - 1, 0, -1):
                self.tree[i] = self.segfunc(self.tree[2 * i], self.tree[2 * i + 1])

        def update(self, k, x):
            """
            k番目の値をxに更新
            k: index(0-index)
            x: update value
            """
            k += self.num
            self.tree[k] = x
            while k > 1:
                self.tree[k >> 1] = self.segfunc(self.tree[k], self.tree[k ^ 1])
                k >>= 1

        def query(self, l, r):
            """
            [l, r)のsegfuncしたものを得る
            l: index(0-index)
            r: index(0-index)
            """
            res = self.ide_ele
            l += self.num
            r += self.num
            while l < r:
                if l & 1:
                    res = self.segfunc(res, self.tree[l])
                    l += 1
                if r & 1:
                    res = self.segfunc(res, self.tree[r - 1])
                l >>= 1
                r >>= 1
            return res

    
    
    
    
    
    
    
    

    def BellmanFord(N, M, ABC, vertex_start, vertex_end, value_if_inf = -1, find_shortest = False):
        """to calculate furthest or shortest length between vertex_start and vertex_end using BellmanFord algorithm
        
        Args:
            N (int): number of vertices
            M (int): number of edges
            ABC (list): [(ai, bi, ci) for _ in range(N)] where i-th edge is directed from vertex ai to vertex bi and the length is ci 
            vertex_start (int): start vertex. usually use 0.
            vertex_end (int): end vertex. usually use N-1.
            value_if_inf (int or string as you like, optional): value you want when the furthest (or shortest) distance is infinite (or -infinite). Defaults to -1.
            find_shortest (bool, optional): choose False to find furthest path. Defaults to False.
        
        Returns:
            int or string: normally int (but can be str if you set value_if_inf to str)
            
        Example:
                N, M, P = R()
                ABC = [R() for _ in range(M)]
                ABC = [(a-1, b-1, c-P) for a, b, c in ABC]
                print(BellmanFord(N, M, ABC, 0, N-1, value_if_inf = 'inf'))

        """

        def make_reachable_list(N, M, ABC, vertex_start, vertex_end):
            reachable_to_direct = defaultdict(list)
            reachable_from_direct = defaultdict(list)
            reachable_from_start = [False] * N
            reachable_to_end = [False] * N
            reachable_from_start[vertex_start] = True
            reachable_to_end[vertex_end] = True
            reachable_from_both_sides = [False] * N
            dfs_from_start = []
            dfs_to_end = []
            for a, b, c in ABC:
                reachable_to_direct[a].append(b)
                reachable_from_direct[b].append(a)
                if a == vertex_start:
                    dfs_from_start.append(b)
                    reachable_from_start[b] = True
                if b == vertex_end:
                    dfs_to_end.append(a)
                    reachable_to_end[a] = True
            while dfs_from_start:
                v = dfs_from_start.pop()
                for i in reachable_to_direct[v]:
                    if not reachable_from_start[i]:
                        reachable_from_start[i] = True
                        dfs_from_start.append(i)
            while dfs_to_end:
                v = dfs_to_end.pop()
                for i in reachable_from_direct[v]:
                    if not reachable_to_end[i]:
                        reachable_to_end[i] = True
                        dfs_to_end.append(i)
            for i in range(N):
                reachable_from_both_sides[i] = reachable_from_start[i] and reachable_to_end[i]
            return reachable_from_both_sides
        
        reachable_from_both_sides = make_reachable_list(N, M, ABC, vertex_start, vertex_end)
        
        if find_shortest:
            dist = [INF for i in range(N)]
        else:
            dist = [-INF for i in range(N)]
        
        dist[vertex_start] = 0
        for i in range(N):
            updated = False
            for a, b, c in ABC:
                if not reachable_from_both_sides[a]:
                    continue
                elif find_shortest:
                    update_condition = dist[a] + c < dist[b]
                else:
                    update_condition = dist[a] + c > dist[b]
                if dist[a] != INF and update_condition:
                    dist[b] = dist[a] + c
                    updated = True
                    if i == N-1:
                        return value_if_inf
            if not updated:
                break
        return dist[vertex_end]

    
    def warshall_floyd(number_vertex, XYD, shift_one_for_vertex=False):
        
        
        
        
        
        d = [[float("inf")]*number_vertex for i in range(number_vertex)]
        for x, y, dist in XYD:
            if shift_one_for_vertex:
                x = x - 1
                y = y - 1
            d[x][y] = dist
            d[y][x] = dist
        for i in range(number_vertex):
            d[i][i] = 0 
        for k in range(number_vertex):
            for i in range(number_vertex):
                for j in range(number_vertex):
                    d[i][j] = min(d[i][j],d[i][k] + d[k][j])
        return d


    """ initialize variables and set inputs
    
        
        
            
            
    
        
        
        
        
        
        
        
    
        
        
        
        
    
        
        
    """

    
    r = lambda: sys.stdin.readline().strip()
    r_int = lambda: int(r())
    r_float = lambda: float(r())
    R = lambda: list(map(int, r().split()))
    R_str = lambda: r().split()
    R_map = lambda: map(int, r().split())
    R_float = lambda: list(map(float, r().split()))
    R_tuple = lambda: tuple(map(int, r().split()))

    """ how to treat input
    
    
    
    
    
    
    
    """

    
    if sample_file:
        sys.stdin = open(sample_file)
        
    
    
    
    
    N, X, M = R()
    arrived = [0] * (M+1)
    sums = [0] * (M+1)
        
    現在値 = X
    回数 = 1 
    合計値 = 0 
    if not 現在値:
        print(0)
        return
    while 現在値:
        合計値 += 現在値
        sums[現在値] = 合計値
        arrived[現在値] = 回数

        
        現在値 = (現在値**2)%M
        回数 += 1
        
        
        if arrived[現在値]:
            before_round = arrived[現在値] - 1
            before_round_sum = sums[現在値] - 現在値
            one_round = 回数 - arrived[現在値]
            one_round_sum = 合計値 - sums[現在値] + 現在値
            break
    if not 現在値:
        ans = 0
        for i in range(N):
            ans += X
            X = (X**2)%M
            if not X:
                print(ans)
                return
        print(ans)
        return
    if one_round + before_round >= N:
        ans = 0
        for i in range(N):
            ans += X
            X = (X**2)%M
    else:
        a, b = divmod(N-before_round, one_round)
        ans = before_round_sum
        ans += a * one_round_sum
        for i in range(b):
            ans += 現在値
            現在値 = (現在値**2)%M
    print(ans)

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

    


    
    
    

    """memo: how to use defaultdict of list
    
    Dic = defaultdict(list)
    
    Dic[x].append(y)
    
    for k, v in Dic.items():
    """
    
    """memo: how to solve binary problems
    
    >>> a = 5
    >>> bin_str_a = format(a, '
    >>> print(bin_str_a)
    0b0101 
    """
    
    """memo: how to solve the problem
    creating simple test/answer
    greed
    simple dp
    graph
    """

if __name__ == '__main__':
    main()

Solution (Python):
n, x, m = map(int, input().split())
a = x
dup = [0]*(10**5+10)
mod = [a]
loop = []
cnt = 0
while cnt < n:
    a = a**2 % m
    if dup[a]==1:
        i = mod.index(a)
        before = mod[:i]
        loop = mod[i:]
        break
    mod.append(a)
    dup[a] = 1
    cnt += 1

length = len(loop)
if length == 0:
    print(sum(mod[:n]))
else:
    t = (n-i)//length
    amari = (n-i) % length
    ans = sum(before) + t * sum(loop) + sum(loop[:amari])
    print(ans)

Solution (Python):
n,x, m = map(int, input().split())
mod = [None for _ in range(m)]
count = x
loop = []
rem = [x]
for i in range(1, n):
    value = (x * x) % m
    rem.append(value)
    x = (x * x) % m
    if mod[value] != None:
        
        s_index = mod[value]
        loop = rem[s_index : -1]
        num = (n - s_index) // len(loop)
        amari = (n - s_index) % len(loop)
        if amari == 0:
            print(sum(rem[:s_index]) + sum(loop) * num) 
        else:
            print(sum(rem[:s_index]) + sum(loop) * num + sum(loop[:amari])) 
        exit()
    else:
        mod[value] = i
        count += value
print(count)

Solution (Python):
n, x, m = map(int, input().split())
a = x
S = [0] * (m + 1)
F = [None] * m
F[x] = 0
i0, j0 = None, None
for i in range(1, m + 1):
    S[i] = S[i - 1] + a
    a = pow(a, 2, m)
    if F[a] is not None:
        i0, j0 = F[a], i
    F[a] = i
q, r = (n - i0) // (j0 - i0), (n - i0) % (j0 - i0)
print((S[j0] - S[i0]) * q + S[i0 + r])

Solution (Python):
N, x, M = [int(a) for a in input().split()]


g = [x] + [0] * M
ginv = [-1] * (M + 1)
ginv[x] = 0
partSum = [x] + [0] * (M + 1)     

for i in range(1, M + 1):
    g[i] = g[i - 1] * g[i - 1] % M
    if ginv[g[i]] >= 0:  
        iStart = ginv[g[i]]
        cycleSum = partSum[i - 1] - partSum[iStart - 1]
        cycleLen = i - iStart

        numCycle = (N - 1 - (i - 1)) // cycleLen
        remCycle = (N - 1 - (i - 1)) % cycleLen
        ans = partSum[i - 1] + cycleSum * numCycle + partSum[iStart + remCycle - 1] - partSum[iStart - 1]
        break
    else:
        ginv[g[i]] = i
        partSum[i] = partSum[i - 1] + g[i]

print(ans)

Solution (Python):


def main():
    n, x, m = map(int, input().split())
    mflg = [-1] * m
    mflg[x] = 0
    ltot = [x]

    tot = x
    a = x
    for i in range(1, n):
        a = a ** 2 % m

        if mflg[a] >= 0:
            b = mflg[a]
            z = (n - i) // (i - b)
            y = (n - i) % (i - b)
            if b == 0 and y == 0:
                tot += ltot[i - 1] * z
            elif b == 0:
                tot += ltot[b - 1 + y] + ltot[i - 1] * z
            else:
                tot += (ltot[b - 1 + y] - ltot[b - 1]) + (ltot[i - 1] - ltot[b - 1]) * z
            break

        mflg[a] = i
        tot += a
        ltot.append(tot)

    print(tot)


main()

Solution (Python):
n, x, m = (int(x) for x in input().split())
dic = {}
count = 1
dic[x]=0
ans = [x]
bool = True
while (count < n):
    x = pow(x, 2, m)
    if x not in dic:
        dic[x] = count
        ans.append(x)
    else:
        roop = ans[dic[x]:count]
        ans = sum(ans[:dic[x]])
        roop_count = (n-dic[x])//len(roop)
        roop_hasuu = (n-dic[x])%len(roop)
        ans+=roop_count*sum(roop)+sum(roop[:roop_hasuu])
        print(ans)
        bool = False
        count = pow(10, 13)
    count+=1
if bool:
    print(sum(ans))

Solution (Python):
n,x,m = map(int,input().split())
set_a = set()
a = []
while len(a)<n:
    if x in a:break
    set_a.add(x)
    a.append(x)
    x = x**2%m

ans = sum(a[:min(n,len(a))])
n -= len(a)
if n<1:
    print(ans)
    exit()

i = a.index(x)
l = len(a)-i
ans += sum(a[i:]) * (n//l)
n %= l
ans += sum(a[i:i+n])
print(ans)

Solution (Python):
import numpy as np
i8 = np.int64


def solve(N, X, M):
    memo_val = np.zeros(M, i8)
    memo_i = np.zeros(M, i8)
    ret = 0
    a = X
    for i in range(1, N):
        ret += a
        if memo_i[a] > 0:
            u = (N - memo_i[a]) // (i - memo_i[a])
            v = (N - memo_i[a]) % (i - memo_i[a])
            ret = u * (ret - memo_val[a])
            nokori = v + memo_i[a]
            for j in range(M):
                if memo_i[j] == nokori:
                    ret += memo_val[j]
            return ret
        memo_i[a] = i
        memo_val[a] = ret
        a = a ** 2 % M
    ret += a
    return ret

def main():
    f = open(0)
    N, X, M = [int(x) for x in f.readline().split()]
    ans = solve(N, X, M)
    print(ans)

main()

Solution (Python):
n,x,m = map(int,input().split())
mod = m
amari = [-1]*m
a = x
cnt = 1
l = 0
r = 0
while 1:
    
    if cnt == n:
        break
    
    if amari[a] != -1:
        l = amari[a]
        r = cnt
        
        break
    else:
        amari[a] = cnt
    cnt += 1
    a = ((a % m)**2) % m
    
    
    
ans = 0
if cnt == n:
    l = 0
    r = n

a = x
temp = 0

for i in range(l-1):
    ans += a
    a = (a**2) % mod



for i in range(r-l):
    temp += a
    a = (a**2) % mod
ans += temp * ((n-l+1)//(r-l))



if r != n:
    for i in range((n-l+1)%(r-l)):
        ans += a
        a = (a**2) % mod
        
if n == 1:
    ans //=2
print(ans)

Solution (Python):
n, x, m = map(int, input().split())
l = [x]
a = x
loopsum = 0
loopnum = 0


for i in range(n - 1):
    a = a**2%m
    if a not in l:
        l.append(a)
    else:
        pos = l.index(a)
        loopnum = len(l[pos:])
        loopsum = sum(l[pos:])
        break

if loopnum > 0:
    rest = (n - len(l))%loopnum
    times = (n - len(l))//loopnum
    ans = sum(l) + loopsum*times
    for i in range(rest):
        ans += l[pos + i]
else:
    ans = sum(l)

print(ans)

Solution (Python):
from collections import defaultdict
n, x, m = list(map(int, input().split()))
dp = [False]*(m+1)
a = [0]*(m+1)
d = defaultdict(lambda: set())
flag = False
for i in range(1, min(m+1, n+1)):
    if i == 1:
        a[i] = x
        dp[x] = True
    else:
        a[i] = a[i-1]**2 % m
        if not dp[a[i]]:
            d[a[i-1]] = a[i]
            dp[a[i]] = True

        else:
            startv = a[i]
            flag = True

            break
start_idx = a[1:].index(a[i])+1
loop_haba = i-start_idx
loop_sum = sum(a[start_idx:i])
if not flag:
    print(sum(a[:n+1]))
else:
    ans = sum(a[:(start_idx)])

    ans += ((n-start_idx)//loop_haba)*loop_sum
    ans += sum(a[start_idx: start_idx+((n-start_idx) % loop_haba)+1])
    print(ans)

Solution (Python):
""" E
    https://atcoder.jp/contests/abc173/tasks/abc173_e
"""
import sys
import math
from functools import reduce

def readString():
    return sys.stdin.readline()

def readInteger():
    return int(readString())

def readStringSet(n):
    return sys.stdin.readline().split(" ")[:n]

def readIntegerSet(n):
    return list(map(int, readStringSet(n)))

def readIntegerMatrix(n, m):
    return reduce(lambda acc, _: acc + [readIntegerSet(m)], range(0, n), [])

def main(N, X, M):
    A = X
    m = {A: A}
    i = -1
    for j in range(1, M):
        A = (A * A) % M if M > 0 else 0
        if A in m:
            i = list(m.keys()).index(A)
            break
        else:
            m[A] = A

    if i == -1:
        return sum(m.values())
    else:
        l = list(m.values())
        s1 = sum(l[:i])
        len_repeat = len(l) - i
        s2 = sum(l[i:])

        return s1 + int((N-i)/len_repeat) * s2 + sum(l[i:(N-i)%len_repeat+i])

if __name__ == "__main__":
    _N, _X, _M = readIntegerSet(3)

    print(main(_N, _X, _M))

Solution (Python):
n, x, m = (int(x) for x in input().split())
A = [x]
used = {x, }
while True:
    x = pow(x, 2, m)
    if x in used:
        index = A.index(x)
        break
    else:
        A.append(x)
        used.add(x)
if n <= len(A):
    print(sum(A[:n]))
else:
    ans = sum(A)
    n -= len(A)
    m = len(A) - index
    ans += (n // m) * sum(A[index:])
    ans += sum(A[index:index + (n % m)])
    print(ans)

Solution (Python):
n,x,m=map(int,input().split())
tab=[0 for i in range(10**5+1)]

end=False
now=x
for i in range(1,n+1):
    if tab[now]>0:
        las=tab[now]
        cyc=i-las
        break
    tab[now]=i
    now=now*now%m
else:
    end=True
    
if end:
    ans=0
    for i in range(10**5+1):
        if tab[i]>0:
            ans+=i
    print(ans)
else:
    ans=0
    head=0
    body=0
    foot=0
    lop=(n-las)//cyc
    rem=(n-las)%cyc
    for i in range(10**5+1):
        if tab[i]>=las and las+rem>=tab[i]:
            body+=i
            foot+=i
        elif tab[i]>=las:
            body+=i
        elif tab[i]>0:
            head+=i
    ans=head+body*lop+foot
    print(ans)

Solution (Python):


def solve():
    N, X, M = list(map(int, input().split()))

    def f(x, m):
        return x%m

    
    
    
    
    A = [0]*(10**5+2)  
    Ar = [0]*(10**5+2)  
    A[0], Ar[0] = X, X
    Adic = {X:0}  
    stop_i = 0
    is_zero = False
    for i in range(1, N):
        A[i] = f(pow(A[i-1], 2, M), M)
        Ar[i] = Ar[i-1] + A[i]
        if A[i] == 0:
            is_zero = True
            stop_i = i
            break
        if A[i] in Adic:
            stop_i = i
            break
        Adic[A[i]] = i

    if stop_i == 0:
        print(Ar[N-1])
    elif is_zero:
        print(Ar[stop_i])
    else:
        start_i = Adic[A[stop_i]]
        t = stop_i - start_i
        restN = N - start_i
        if start_i-1 >= 0:
            ans = Ar[start_i-1]
            ans += (Ar[stop_i-1] - Ar[start_i-1]) * (restN//t)
            ans += Ar[start_i-1+(restN%t)] - Ar[start_i-1]
        else:
            ans = Ar[stop_i-1] * (restN//t)
            ans += Ar[start_i-1+(restN%t)]
        print(ans)


if __name__ == "__main__":
    solve()

Solution (Python):
n, x, m = map(int, input().split())

def solve(n, x, m):
    if n == 1:
        return x
    arr = [x]
    for i in range(1, n):
        x = x*x % m
        if x in arr:
            rem = n-i
            break
        else:
            arr.append(x)
    else:
        rem = 0
    sa = sum(arr)
    argi = arr.index(x)
    roop = arr[argi:]
    nn, r = divmod(rem, len(roop))
    return sa + nn*sum(roop) + sum(roop[:r])

print(solve(n, x, m))

Solution (Python):




import sys









MOD = 10 ** 9 + 7
INF = 10 ** 9
PI = 3.14159265358979323846

def read_str():      return sys.stdin.readline().strip()
def read_int():      return int(sys.stdin.readline().strip())
def read_ints():     return map(int, sys.stdin.readline().strip().split())
def read_ints2(x):   return map(lambda num: int(num) - x, sys.stdin.readline().strip().split())
def read_str_list(): return list(sys.stdin.readline().strip().split())
def read_int_list(): return list(map(int, sys.stdin.readline().strip().split()))
def GCD(a: int, b: int) -> int: return b if a%b==0 else GCD(b, a%b)
def LCM(a: int, b: int) -> int: return (a * b) // GCD(a, b)

def Main():
    n, x, m = read_ints()
    query = []
    s = set([])
    while x not in s:
        s.add(x)
        query.append(x)
        x = x * x % m
    
    frm = query.index(x)
    length = len(query) - frm
    total = sum(query[frm:])
    
    ans = sum(query[:frm]) + total * ((n - frm) // length) + sum(query[frm:frm + (n - frm) % length])
    print(ans)

if __name__ == '__main__':
    Main()

Solution (Python):
N, X, M = map(int, input().split())
A = [X]
A_dict = {X}
a = X
s = X
r = 0
r0 = 0
l = 0
amari = 0

for i in range(M+1):
  a = a ** 2 % M
  if a in A_dict:
    r0 = A.index(a)
    l = len(A[r0:])
    r = (N - i - 1) // l
    amari = (N - i - 1) % l
    s = s + sum(A[r0:])*r + sum(A[r0:r0+amari])
    break
  A.append(a)
  A_dict.add(a)
  s += A[-1]

print(s)