Problem:
 Score : 400 points    Problem Statement Given is a string S . Each character in S is either a digit ( 0 , ..., 9 ) or ? .  Among the integers obtained by replacing each occurrence of ? with a digit, how many have a remainder of 5 when divided by 13 ? An integer may begin with 0 .  Since the answer can be enormous, print the count modulo 10^9+7 .      Constraints   S is a string consisting of digits ( 0 , ..., 9 ) and ? .  1 \leq |S| \leq 10^5         Input  Input is given from Standard Input in the following format:  S       Output  Print the number of integers satisfying the condition, modulo 10^9+7 .        Sample Input 1 ??2??5      Sample Output 1 768  For example, 482305, 002865, and 972665 satisfy the condition.       Sample Input 2 ?44      Sample Output 2 1  Only 044 satisfies the condition.       Sample Input 3 7?4      Sample Output 3 0  We may not be able to produce an integer satisfying the condition.       Sample Input 4 ?6?42???8??2??06243????9??3???7258??5??7???????774????4?1??17???9?5?70???76???      Sample Output 4 153716888

Solution (Python):
p = 10**9+7
S = input().strip()
n = len(S)
dp = [[0 for _ in range(13)] for _ in range(n+1)]
dp[0][0] = 1
for i in range(n):
    for j in range(13):
        if S[i]!="?":
            dp[i+1][(10*j+int(S[i]))%13] = dp[i][j]
        else:
            for k in range(10):
                dp[i+1][(10*j+k)%13] = (dp[i+1][(10*j+k)%13]+dp[i][j])%p
print(dp[n][5])

Solution (Python):

MOD = 10**9+7
s = input()
n = len(s)

dp = [[0]*13 for _ in range(n+1)]
dp[0][0] = 1
b = 1
for i in range(1,n+1):
    if s[n-i] == "?":
        for j in range(10):
            for k in range(13):
                dp[i][(j*b+k)%13] += dp[i-1][k]
    else:
        j = int(s[n-i])
        for k in range(13):
            dp[i][(j*b+k)%13] += dp[i-1][k]
    for k in range(13):
        dp[i][k] %= MOD
    b *= 10
    b %= 13

print(dp[n][5]%MOD)

Solution (Python):
rotate = lambda x,y: x[-y:] + x[:-y]
S = input()
mod = 13
amari = 5
dp_tmp = [0]*13
dp_tmp[0] = 1
mul = 1
for s in S[::-1]:
    dp = [0]*13
    if s == "?":
        for n in range(10):
            dp_tmp_roll = rotate(dp_tmp,n*mul%mod)
            for k in range(13):
                dp[k] += dp_tmp_roll[k]
    else:
        dp_tmp_roll = rotate(dp_tmp,int(s)*mul%mod)
        for k in range(13):
            dp[k] += dp_tmp_roll[k]
    for n in range(13):
        dp[n] = dp[n]%(10**9+7)
    dp_tmp = dp
    mul = mul*10%13
print(dp[amari])

Solution (Python):
s = input()
q = s.count("?")
ans = [0] * 13
ans[0] = 1

for v in s:
    dp = [0] * 13
    for j in range(13):
        dp[(j * 10) % 13] = ans[j] % 1000000007
    dp += dp
    for j in range(13):
        if v == '?':
            ans[j] = sum(dp[j+4:j+14])
        else:
            ans[j] = dp[j - int(v)]

print(ans[5] % 1000000007)

Solution (Python):
tdlist = lambda i, j, value=0: [[value] * j for _ in range(i)]
MOD = 10**9+7

s = [-1 if x == "?" else int(x) for x in input()[::-1]]
n = len(s)
m = 13
dp = tdlist(n+1, m)
dp[0][0] = 1

digrem = 1
for i in range(n):
    adds = [digrem * j % m for j in range(10)]
    for j in range(m):
        if not dp[i][j]: continue
        if s[i] == -1:
            for k in range(10):
                dp[i+1][(j+adds[k]) % m] += dp[i][j]  % MOD     
        else:
            dp[i+1][(j+adds[s[i]]) % m] += dp[i][j] % MOD

    digrem = digrem * 10 % m

print(dp[n][5] % MOD)

Solution (Python):
def main():
  s = input()
  n = len(s)
  mod = 10**9+7
  dp = [[0]*13 for _ in range(n+1)]
  dp[0][0] = 1
  d = {(j, k):(10*j+k)%13 for k in range(10) for j in range(13)}
  for i in range(n):
    if s[i] != "?":
      t = int(s[i])
      for j in range(13):
        dp[i+1][d[(j, t)]] += dp[i][j]
        dp[i+1][d[(j, t)]] %= mod
    else:
      for k in range(10):
        for j in range(13):
          dp[i+1][d[(j, k)]] += dp[i][j]
          dp[i+1][d[(j, k)]] %= mod
  print(dp[n][5]%mod)
if __name__ == '__main__':
  main()

Solution (Python):
S = list(input())
dp = [[0]*13 for j in range(len(S)+1)]
dp[0][0] = 1
mod = 10**9 + 7
k = 1
for i, s in enumerate(S[::-1]):
    if s == '?':
        for c in range(10):
            for j in range(13):
                dp[i+1][(k*c+j) % 13] += dp[i][j] % mod
    else:
        c = int(s)
        for j in range(13):
            dp[i+1][(k*c+j) % 13] += dp[i][j] % mod
    k = k * 10 % 13
print(dp[-1][5] % mod)

Solution (Python):
S=input()
l=len(S)
mod=10**9+7
dp=[[0 for j in range(13)] for i in range(l+1)]
dp[0][0]=1

for i in range(l):
  if S[i]=='?':
    for k in range(13): 
      for j in range(10):      
        dp[i+1][(k*10+j)%13]+=dp[i][k]%mod
  else:
    for k in range(13):
      dp[i+1][(k*10 + int(S[i]))%13]+=dp[i][k]%mod
      
print(dp[-1][5]%mod)

Solution (Python):
import numpy as np


def main():
    S = {i:s for i, s in enumerate(input()[::-1])}
    MOD = 10**9 + 7
    
    DP_table = np.zeros(13).astype(np.int64)
    
    
    digit = S[0]
    if digit == '?':
        for i in range(10):
            DP_table[i] = 1
    else:
        DP_table[int(digit)] = 1
        
    transition_matrixes = {}
    for multiplier in range(13):
        for digit in range(10):
            transition_matrix = np.zeros((13, 13)).astype(np.int64)
            for k in range(13):
                transition_matrix[(digit * multiplier + k) % 13, k] = 1
            transition_matrixes[(multiplier, digit)] = transition_matrix

    transition_matrixes2 = {}
    for multiplier in range(13):
        transition_matrixes2[multiplier] = np.zeros((13, 13)).astype(np.int64)
        for digit in range(10):
            transition_matrixes2[multiplier] += transition_matrixes[(multiplier, digit)]
            
    
    multiplier = 1
    for i in range(1, len(S)):
        multiplier = multiplier * 10 % 13
        digit = S[i]
        if digit == '?':
            transition_matrix = transition_matrixes2[multiplier]
        else:
            transition_matrix = transition_matrixes[(multiplier, int(digit))]
        
        DP_table = np.dot(transition_matrix, DP_table)
        DP_table %= MOD

    print(int(DP_table[5]))

    
if __name__ == "__main__":
    main()

Solution (Python):
mod = 10 ** 9 + 7

S = input()

dp = [[0] * 13 for _ in range(len(S) + 1)]
dp[0][0] = 1
mul = 1

for i in range(len(S)):
    x = S[-(i + 1)]
    if x == "?":
        for k in range(10):
            for j in range(13):
                dp[i + 1][(mul * k + j) % 13] += dp[i][j]
                dp[i + 1][(mul * k + j) % 13] %= mod
    else:
        k = int(x)
        for j in range(13):
            dp[i + 1][(mul * k + j) % 13] += dp[i][j]
            dp[i + 1][(mul * k + j) % 13] %= mod
    mul = mul * 10 % 13  

print(dp[-1][5])

Solution (Python):
def main():
    S = {i:s for i, s in enumerate(input()[::-1])}
    MOD = 10**9 + 7

    DP_table = [[0 for i in range(13)] for j in range(len(S))]
    
    digit = S[0]
    if digit == '?':
        for i in range(10):
            DP_table[0][i] = 1
    else:
        DP_table[0][int(digit)] = 1
    
    multiplier = 1
    for i in range(1, len(S)):
        digit = S[i]
        multiplier = multiplier * 10 % 13
        if digit == '?':
            for j in range(10):
                for k in range(13):
                    DP_table[i][(j * multiplier + k) % 13] += DP_table[i-1][k]
        else:
            digit = int(digit)
            for k in range(13):
                DP_table[i][(digit * multiplier + k) % 13] += DP_table[i-1][k]
        
        for j in range(13):
            DP_table[i][j] %= MOD

    print(DP_table[len(S)-1][5])

    
if __name__ == "__main__":
    main()

Solution (Python):
import sys
input = sys.stdin.readline


def main():
    mod = 10**9+7
    S = input().strip()
    N = len(S)
    dp = [[0]*13 for _ in range(N+1)]
    dp[0][0] = 1
    m = 1
    for n in range(N):
        c = S[N-n-1]
        for i in range(10):
            if c == "?" or int(c) == i:
                for j in range(13):
                    dp[n+1][(j + i*m) % 13] += dp[n][j] % mod
        m = m * 10 % 13
    print(dp[N][5] % mod)


if __name__ == '__main__':
    main()

Solution (Python):
s = input()
mod = 10**9+7

init = [0 for i in range(13)]
if s[0] == '?':
  dp = [1] * 10 + [0] * 3
else:
  dp = [0 for i in range(13)]
  dp[int(s[0])] = 1

for c in s[1:]:
  d = [0 for i in range(13)]
  if c == '?':
    for i in range(13):
      for j in range(10):
        idx = (i * 10 + j) % 13
        d[idx] = (dp[i] + d[idx]) % mod 
  else :
    for i in range(13):
      idx = (i * 10 + int(c)) % 13
      d[idx] = (dp[i] + d[idx]) % mod
  dp = d
print(dp[5])

Solution (Python):
s = input()
n = len(s)
mod = 10**9+7
dp = [[0]*13 for _ in range(n+1)]
dp[0][0] = 1
d = {(j, k):(10*j+k)%13 for k in range(10) for j in range(13)}
for i in range(n):
  if s[i] != "?":
    t = int(s[i])
    for j in range(13):
      dp[i+1][d[(j, t)]] += dp[i][j]
      dp[i+1][d[(j, t)]] %= mod
  else:
    for k in range(10):
      for j in range(13):
        dp[i+1][d[(j, k)]] += dp[i][j]
        dp[i+1][d[(j, k)]] %= mod
print(dp[n][5]%mod)

Solution (Python):
s=input()
n=len(s)
p=10**9+7
 
dp=[[0]*13 for i in range(n)]
m=[0]*(n-1)+[1]
for i in range(n-1):
    m[-i-2]=m[-i-1]*10%13
 
if s[0]!='?':
    a=int(s[0])*m[0]%13
    dp[0][a]=1
else:
    for k in range(10):
        a=k*m[0]%13
        dp[0][a]=1
 
for i in range(1,n):
    if s[i]!='?':
        for j in range(13):
            dp[i][j]=dp[i-1][(j-int(s[i])*m[i])%13]%p
    else:
        for k in range(10):
            for j in range(13):
                dp[i][j]+=dp[i-1][(j-k*m[i])%13]%p
print(dp[n-1][5]%p)

Solution (Python):
S = input()
n = len(S)
mod = 10**9 + 7
pow10 = [1]
for i in range(1,n):
    x = pow10[-1]
    y = 10*x%13
    pow10.append(y)
dp = [[0 for i in range(13)]for j in range(n+1)]
dp[0][0] = 1
for i in range(n):
    d = S[-i-1]
    if d == '?':
        for j in range(10):
            s = (pow10[i]*j)%13
            for t in range(13):
                r = (s+t)%13
                dp[i+1][r] = (dp[i+1][r] + dp[i][t])%mod
    else:
        j = int(d)
        s = (pow10[i]*j)%13
        for t in range(13):
            r = (s+t)%13
            dp[i+1][r] = (dp[i+1][r] + dp[i][t])%mod
print(dp[n][5])

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

    """ convenient functions
    
    
    
    
    
    """

    """convenient decorator
    
    
        
        
        
        
        
        
        
        
        
        
    """
    
    
    import sys
    sys.setrecursionlimit(10**7)
    from itertools import accumulate, combinations, permutations, product 
    
    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 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)
    
    
    
    INF = 10 ** 18
    MOD = 10 ** 9 + 7
    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
        >>  else:
        >>      do something
        >>      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]

    
    
    
    
    

    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]

    """ 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_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)
        
    
    

    S = r()
    N = len(S)
    rem = 1 
    dp = [[0]*(N+1) for _ in range(13)]
    dp[0][0] = 1
    for i, s in enumerate(reversed(S)):
        if s == '?':
            for j in range(10):
                d = (rem*j)%13
                for k in range(13):
                    dp[(k+d)%13][i+1] = (dp[(k+d)%13][i+1] + dp[k][i]) % (10**9+7)
        else:
            ss = int(s)
            d = (rem*ss)%13
            for k in range(13):
                dp[(k+d)%13][i+1] = dp[k][i]
        
        rem = rem*10%13
    print(dp[5][N])
        
    
    
    
    

    """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):
S = input()

dp = [[0,0,0,0,0,0,0,0,0,0,0,0,0] for i in range(len(S) + 2)]
INF = 10**9 + 7
for num, dig in enumerate(S):
    if num == 0:
        if dig == '?':
            dp[num] = [1,1,1,1,1,1,1,1,1,1,0,0,0]
        else:
            dp[num] = [0,0,0,0,0,0,0,0,0,0,0,0,0]
            dp[num][int(dig)] = 1

    elif dig == '?':
        for j in range(10):
            for i in range(13):
                dp[num][(i * 10 + j) % 13] += dp[num - 1][i] % INF
                dp[num][(i * 10 + j) % 13] %= INF

    else:
        for i in range(13):
            dp[num][(i*10 + int(dig))%13] += dp[num - 1][i] % INF
            dp[num][(i * 10 + int(dig)) % 13] %= INF

print(dp[len(S)-1][5])

Solution (Python):
import sys
input = sys.stdin.readline
sys.setrecursionlimit(10 ** 7)
MOD = 10**9+7

S = input().rstrip("\n")



dp = [[0]*13 for _ in range(len(S)+1)]


if S[-1] == "?":
    dp[1][:10] = [1]*10
else:
    dp[1][int(S[-1])] += 1


mul = 10

for i in range(2,len(S)+1):
    if S[-i] == "?":
        
        for j in range(10):
            jmod = j*mul % 13
            
            for k,x in enumerate(dp[i-1]):
                dp[i][(k+jmod)%13] = (dp[i][(k+jmod)%13] + x) % MOD
    else:
        jmod = int(S[-i])*mul % 13
        for k,x in enumerate(dp[i-1]):
            dp[i][(k+jmod)%13] = (dp[i][(k+jmod)%13] + x) % MOD
    mul = (mul * 10) % 13

print(dp[-1][5])

Solution (Python):
import numpy as np

S = input()
mod = int(10**9 + 7)
dp = np.zeros(13, dtype=int)
dp[0] = 1

M = np.zeros((13, 13), dtype=int)
for i in range(13):
    for d in range(10):
        M[i][(i * 10 + d) % 13] += 1

for s in S:
    ndp = np.zeros(13, dtype=int)
    if s == '?':
        ndp = np.dot(dp, M)
    else:
        d = int(s)
        for j in range(13):
            ndp[(j * 10 + d) % 13] += dp[j]
    dp = ndp % mod

print(dp[5])

Solution (Python):
import numpy as np

s = input()
p = np.array([0]*13)
p[0] = 1

amari = [[0]*13 for _ in range(13)]
for j in range(10):
  for i in range(13):
    amari[i][(i*10+j)%13] += 1
M = np.array(amari)

for si in range(len(s)):
  a = np.array([0]*13)
  if s[si] == '?':
    a = np.dot(p,M)
  else:
    c = int(s[si])
    for i in range(13):
      a[(i*10+c)%13] += p[i]
  p = np.mod(a,1000000007)
print(p[5])

Solution (Python):
import numpy as np
 
s = input()
p = np.array([0]*13)
p[0] = 1
 
amari = [[0]*13 for _ in range(13)]
for j in range(10):
  for i in range(13):
    amari[i][(i*10+j)%13] += 1
M = np.array(amari)
 
for si in range(len(s)):
  a = np.array([0]*13)
  if s[si] == '?':
    a = np.dot(p,M)
  else:
    c = int(s[si])
    for i in range(13):
      a[(i*10+c)%13] += p[i]
  p = np.mod(a,1000000007)
print(p[5])

Solution (Python):
def digits_parade(S: str)->int:
    MOD = 10**9 + 7
    DIV = 13
    
    dp = [[0] * DIV for _ in range(len(S)+1)]
    dp[0][0] = 1

    
    digit = 1

    for i, c in enumerate(reversed(S)):
        if c == '?':
            for n in range(10):
                d = (n * digit) % DIV
                for j in range(DIV):
                    dp[i+1][(j+d) % DIV] += dp[i][j]
                    dp[i+1][(j+d) % DIV] %= MOD
        else:
            d = (int(c) * digit) % DIV
            for j in range(DIV):
                dp[i+1][(j+d) % DIV] += dp[i][j]
                dp[i+1][(j+d) % DIV] %= MOD

        digit = (digit * 10) % DIV

    return dp[len(S)][5]


if __name__ == "__main__":
    S = input()
    ans = digits_parade(S)
    print(ans)

Solution (Python):
import sys

if sys.platform =='ios':
	sys.stdin=open('Untitled.txt')
input = sys.stdin.readline
def INT(): return int(input())
def MAP(): return [int(s) for s in input().split()]

S = input().rstrip()

dp = [[0] * 13 for _ in range(len(S)+1)]
dp[0][0] = 1



for i in range(len(S)):
	for j in range(13):
		if S[i]=='?':
			for d in range(10):
				nj = (j*10+d)%13
				dp[i+1][nj] += dp[i][j]
		else:
			nj = (j*10+int(S[i]))%13
			dp[i+1][nj] += dp[i][j]
	for j in range(13):
		dp[i+1][j] %= (10**9+7)


print(dp[len(S)][5])

Solution (Python):
s=input()
d=[0]*13
d[0]=1
for i in s:
    dnew=[0]*13
    if i=="?":
        for j in range(10):
            for k in range(13):
                dnew[(10*k+j)%13]+=d[k]
            for k in range(13):
                dnew[k]%=(10**9+7)
    else:
        for k in range(13):
            dnew[(10*k+int(i))%13]+=d[k]
        for k in range(13):
            dnew[k]%=(10**9+7)
    d=dnew

print(d[5])

Solution (Python):
import sys

for S in sys.stdin:
    break
S = S.strip()
le = len(S)

mod = 10**9 + 7

mo = 1

dp = ([1]*10 if S[-1] == "?" else [int(x == int(S[-1])) for x in range(10)]) + [0]*3

i = 1
while i != le:
    mo = (mo * 10)%13
    a = [0]*13
    if S[-i-1] == "?":
        j = 0
        while j != 10:
            k = 0
            while k != 13:
                a[(k+mo*j)%13] += dp[k]
                a[(k+mo*j)%13] %= mod
                k += 1
            j += 1
    else:
        j = int(S[-i-1])
        k = 0
        while k != 13:
            a[(k+mo*j)%13] += dp[k]
            a[(k+mo*j)%13] %= mod
            k += 1
    dp = a
    i += 1

print(dp[5])

Solution (Python):
s = input()

N = 13
dp = [0] * N
dp[0] = 1

mod = 10 ** 9 + 7
mul = 1
t = s[::-1]

for i in range(len(s)):
    nextDP = [0] * N
    c = t[i]
    if c == "?":
        for k in range(10):
            for j in range(N):
                nextDP[(k * mul + j) % N] += dp[j]
                nextDP[(k * mul + j) % N] %= mod
    else:
        k = int(t[i])
        for j in range(N):
            nextDP[(k * mul + j) % N] += dp[j]
            nextDP[(k * mul + j) % N] %= mod

    mul *= 10
    mul %= N
    dp = nextDP

print(dp[5])

Solution (Python):

def main():
    s=input()
    n=len(s)
    mod=10**9+7
    
    if s[-1]=="?":
        dp=[1,1,1,1,1,1,1,1,1,1,0,0,0]
    else:
        dp=[0]*13
        dp[int(s[-1])]=1
    
    for i in range(1,n):
        imod=10**(i %12)
        ss0=s[n-i-1]
        if ss0!="?":
            dpnew = [ dp[(j-int(ss0)*imod)%13] %mod for j in range(13)]
        else:
            dpnew = [sum( dp[(j-k*imod)%13] for k in range(10)) %mod for j in range(13) ]
        dp=dpnew[:]
    print(dp[5])
    
if __name__ == '__main__':
    main()

Solution (Python):
MOD = 10**9 + 7

def solve(s, p=13):
    a = [0] * p
    a[0] = 1
    for c in s:
        b = [0] * p
        for i in range(p):
            b[(i*10)%p] = a[i] % MOD
        b += b
        if c == "?":
            for i in range(p):
                a[i] = sum(b[i+4:i+p+1])
        else:
            for i in range(p):
                a[i] = b[i+p-int(c)]
    return a[5] % MOD

s = input()
print(solve(s))

Solution (Python):

import math
import heapq
import bisect

MOD = 10**9+7

S = input()
ans = [1]+[0]*12
 
for c in S:
    dp=[0]*13
    for i in range(13):
        
        dp[(i*10)%13]=ans[i]%MOD
    
    dp+=dp
 
    if c=='?':
        for i in range(13):
            
            ans[i]=sum(dp[i+4:i+14])
    else:
        for i in range(13):
            ans[i]=dp[i+13-int(c)]

print(ans[5]%MOD)

Solution (Python):

import math
import heapq
import bisect

MOD = 10**9+7

S = input()
ans = [0]*13
ans[0]=1
 
for c in S:
    dp=[0]*13
    for i in range(13):
        
        dp[(i*10)%13]=ans[i]%MOD
    
    dp+=dp
 
    if c=='?':
        for i in range(13):
            
            ans[i]=sum(dp[i+4:i+14])
    else:
        for i in range(13):
            ans[i]=dp[i+13-int(c)]

print(ans[5]%MOD)

Solution (Python):
s = input()

s = s[::-1]
mod = 7 + 10**9
keta = 1
dp = [[0 for i in range(13)] for j in range(len(s) + 1)]
dp[0][0] = 1
for i in range(len(s)):
    if s[i] != "?":
        k = (int(s[i]) * keta) % 13
        for j in range(13):
            dp[i + 1][(j + k) % 13] = dp[i][j]
    else:
        for j in range(13):
            for k in range(10):
                l = ( k * keta ) % 13
                dp[i + 1][(j + l) % 13] += dp[i][j] % mod
    keta = keta *10 % 13 
print(dp[-1][5] % mod)

Solution (Python):
MOD = 10**9+7
S = input()
LenS = len(S)
DP = [[0] * 13 for _ in range(LenS+1)]

c = S[0]
if c == '?':
    for i in range(10):
        DP[1][i] = 1
else:
    n = int(c)
    DP[1][n] = 1

for i in range(2,LenS+1):
    c = S[i-1]
    if c == '?':
        for n in range(10):
            for j in range(13):
                k = (10*j+n) % 13
                DP[i][k] += DP[i-1][j] 
                DP[i][k] %= MOD
    else:
        n = int(c)  
        for j in range(13):
            k = (10*j+n) % 13
            DP[i][k] = DP[i-1][j] 
            DP[i][k] %= MOD

print(DP[LenS][5])