**Arithmetic Expressions**

### Problem Statement :

5-year-old Shinchan had just started learning mathematics. Meanwhile, one of his studious classmates, Kazama, had already written a basic calculator which supports only three operations on integers: multiplication , addition , and subtraction . Since he had just learned about these operations, he didn't know about operator precedence, and so, in his calculator, all operators had the same precedence and were left-associative. As always, Shinchan started to irritate him with his silly questions. He gave Kazama a list of integers and asked him to insert one of the above operators between each pair of consecutive integers such that the result obtained after feeding the resulting expression in Kazama's calculator is divisible by . At his core, Shinchan is actually a good guy, so he only gave lists of integers for which an answer exists. Can you help Kazama create the required expression? If multiple solutions exist, print any one of them. Input Format The first line contains a single integer denoting the number of elements in the list. The second line contains space-separated integers denoting the elements of the list. Output Format Print a single line containing the required expressoin. You may insert spaces between operators and operands. Note You are not allowed to permute the list. All operators have the same precedence and are left-associative, e.g., is interpreted as Unary plus and minus are not supported, e.g., statements like , , or are invalid.

### Solution :

` ````
Solution in C :
In C :
#include <stdio.h>
#include <string.h>
#include <math.h>
#include <stdlib.h>
int main() {
int n;
scanf("%d", &n);
unsigned char * ops = malloc(n); // '+' - 1, '-' - 2, '*' - 3
memset(ops, 3, n);
int * nums = malloc(n * sizeof(unsigned int));
for (int i = 0; i < n; i ++) {
scanf("%d", &nums[i]);
}
unsigned char t[1000][101][2];
int i;
for (i = 0; i < n; i ++) {
if (0 == i) {
t[i][nums[i]][0] = 1;
t[i][nums[i]][1] = 1;
}
else {
int j;
for (j = 1; j < 101; j ++) {
if (t[i - 1][j][0]) {
int tmp = (j + nums[i]) % 101;
t[i][tmp][0] = j;
t[i][tmp][1] = 1;
if (0 == tmp) {
break;
}
tmp = (101 + j - nums[i]) % 101;
t[i][tmp][0] = j;
t[i][tmp][1] = 2;
if (0 == tmp) {
break;
}
tmp = (j * nums[i]) % 101;
t[i][tmp][0] = j;
t[i][tmp][1] = 3;
if (0 == tmp) {
break;
}
}
}
if (j < 101) {
break;
}
}
}
int p = 0;
for (int j = i; j > 0; j --) {
ops[j] = t[j][p][1];
p = t[j][p][0];
}
for (int i = 0; i < n - 1; i ++) {
printf("%d", nums[i]);
switch (ops[i + 1]) {
case 1:
printf("+");
break;
case 2:
printf("-");
break;
default:
printf("*");
break;
}
}
printf("%d\n", nums[n - 1]);
free(nums);
free(ops);
return 0;
}
```

` ````
Solution in C++ :
In C++ :
#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;
typedef struct {
string opers;
int remainder;
} expr;
int main(void){
int N;
cin >> N; // 2 - 10,000
vector<int> vals(N);
for(int i=0; i<N; i++) cin >> vals[i]; // each 1 - 100
expr init_expr;
init_expr.remainder = -1;
vector<expr> results(101, init_expr);
results[vals[0]].remainder = vals[0];
for(int step=1; step<N; step++){
vector<expr> res2(101, init_expr);
int stepval = vals[step];
for(int j=0; j<=100; j++){
int amount = results[j].remainder;
if (amount==-1) continue;
int ans1 = (amount+stepval)%101;
int ans2 = ((amount-stepval)+101)%101;
int ans3 = (amount*stepval)%101;
res2[ans1].remainder = ans1;
res2[ans2].remainder = ans2;
res2[ans3].remainder = ans3;
res2[ans1].opers = results[j].opers+"+";
res2[ans2].opers = results[j].opers+"-";
res2[ans3].opers = results[j].opers+"*";
}
results.swap(res2);
}
cout << vals[0];
const string& opstring = results[0].opers;
for(int i=1; i<N; i++){
cout << opstring[i-1] << vals[i];
}
cout << endl;
return 0;
}
```

` ````
Solution in Java :
In Java :
import java.io.*;
import java.util.*;
import java.text.*;
import java.math.*;
import java.util.regex.*;
public class Solution {
static void printResult(int[] arr, String result){
for(int i=0; i<arr.length;i++){
System.out.print(arr[i]);
if(i!=arr.length-1){
System.out.print(result.charAt(i));
}
}
}
public static boolean calculation(int index, long cur, String result){
if(cur%101 ==0){
//System.out.println("optimize" + cur);
for(int i=result.length(); i<arr.length-1; i++)
result+="*";
printResult(arr,result);
return true;
}
if(index == arr.length-1){
if(((long)arr[index]*cur)%101 ==0){
result+="*";
printResult(arr,result);
//System.out.println(((long)arr[index]*cur));
return true;
}
else if(((long)arr[index]+cur)%101 ==0){
result+="+";
printResult(arr,result);
//System.out.println(((long)arr[index]+cur));
return true;
}
else if((cur-(long)arr[index])%101 == 0){
result+="-";
printResult(arr,result);
//System.out.println((cur-(long)arr[index]));
return true;
}
else
return false;
}
if(!calculation(index+1, ((long)arr[index]*cur)%101,result+"*")){
if(!calculation(index+1, ((long)arr[index]+cur)%101,result+"+")){
if(!calculation(index+1, ((long)cur-arr[index])%101,result+"-")){
return false;
}
}
}
return true;
}
static int[] arr;
public static void main(String[] args) {
Scanner in = new Scanner(System.in);
int a;
a = in.nextInt();
arr = new int[a];
for(int i=0; i<a; i++){
arr[i] = in.nextInt();
}
calculation(1, arr[0],"");//System.out.println(result);
}
}
```

` ````
Solution in Python :
In Python3 :
import sys
sys.setrecursionlimit(15000)
ops = [lambda x,y: x + y, lambda x,y: x * y, lambda x,y: x - y]
op2s = ['+', '*', '-', '']
def exp(i, value, l):
if i == n:
return l if value % 101 == 0 else None
if len(cache[i]) > 0 and value in cache[i]:
return cache[i][value]
for k in range(3):
if exp(i + 1, ops[k](value, a[i]), l) != None:
l[i - 1] = k
cache[i][value] = l
return l
cache[i][value] = None
return None
n = int(input())
a = [int(x) for x in input().strip().split(' ')]
cache = [{}] * n
l = exp(1, a[0], [3] * n)
for i in range(n):
sys.stdout.write(str(a[i]))
sys.stdout.write(op2s[l[i]])
sys.stdout.flush()
```

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