Revise paste Untitled

#include <iostream>
using namespace std;

string add_two_numbers(string a, string b) {
    if((int) b.size() > (int) a.size()) {
        swap(a, b);
    }
    int carry = 0;
    string res = "";
    int i = (int) a.size() - 1, j = (int) b.size() - 1;
    while(j >= 0) {
        int digit_a = (int) (a[i] - '0');
        int digit_b = (int) (b[j] - '0');
        int sum = digit_a + digit_b + carry;
        if(sum < 10) {
            res += (sum + '0');
        }
        else {
            carry = 1;
            res += ((sum % 10) + '0');
        }
        i--;
        j--;
    }
    while(i >= 0) {
        int digit_a = (int) (a[i] - '0');
        int sum = digit_a + carry;
        if(sum < 10) {
            carry = 0;
            res += (sum + '0');
        }
        else {
            carry = 1;
            res += ((sum % 10) + '0');
        }
        i--;
    }
    if(carry == 1) {
        res += "1";
    }
    reverse(res.begin(), res.end());
    return res;
}
string multiply_two_numbers(string a, string b) {
    vector<int> v((int) a.size() + (int) b.size(), 0);
    int sz_a = (int) a.size();
    int sz_b = (int) b.size();
    for(int i = sz_a - 1; i >= 0; i--) {
        for(int j = sz_b - 1; j >= 0; j--) {
            v[i + j + 1] += (a[i] - '0') * (b[j] - '0');
            
        }
    }
    for(int i = sz_a + sz_b - 1; i >= 0; i--) {
        if(v[i] >= 10) {
            v[i - 1] += v[i] / 10;
            v[i] %= 10;
        }
    }
    string res = "";
    int i = 0;
    while(v[i] == 0) {
        i++;
    }
    i = 0;
    
    while(i < sz_a + sz_b) {
        res += (v[i] + '0');
        i++;
    }
    
    return res;
}
int main() {
    ios_base::sync_with_stdio(false);
    cout << multiply_two_numbers("123", "85");
    return 0;
}