string multiply(string num1, string num2) {

if (num1[0] == '0' || num2[0] == '0') return "0";

int n1(num1.size()), n2(num2.size()), n(n1+n2);

vector<int> r(n,0);

string s(n,'0');

for (int i = 0; i < n1; ++i) {

for (int j = 0; j < n2; ++ j) {

r[i+j+1] += (num1[i] - '0') * (num2[j] - '0');

}

}

for (int i =n -1; i >0; --i)

{

if (r[i] >9)

{

r[i-1] += r[i]/10;

}

s[i] += r[i] %10;

}

s[0] += r[0];

return s[0] =='0'? string(s.begin() +1, s.end()):s;

}

string multiply(string num1, string num2) {

int N = num1.size(), M = num2.size();

string res(N+M, '0');

for (int i = N - 1; i >= 0; --i)

{

int carry = 0;

for (int j = M - 1; j >= 0; --j)

{

int sum = carry + (res[i+j+1] - '0') +

(num1[i] - '0') * (num2[j] - '0');

res[i+j+1] = sum % 10 + '0';

carry = sum / 10;

}

res[i] += carry;

}

while (res.size() > 1 && res[0] == '0')

res.erase(res.begin());

return res;

}