Как я знаю, включилась ли система?

Вы используете неправильные форматы. После включения -Wall в gcc и исправления предупреждений я получаю работающую программу. Вы также пропустили \n в своем ответе printf()

#include <stdio.h>

int main(int argc, char **argv)
{
  int a, b, sum, diff, rem, multi;
  float div;
  char character;
  printf("Choose the character you want to use(+, -, *, /, %%): ");
  scanf("%c", &character);
  switch(character)
    {
    case '+': //will be used for addition.
      printf("Enter the first and second number: ");
      scanf("%d %d", &a, &b);
      sum = a+b;
      printf("The sum of the %d and %d is %d\n", a, b, sum);
      break;
    case '-': //will be used for difference.
      printf("Enter the first and second number: ");
      scanf("%d %d", &a, &b);
      diff = a-b;
      printf("The difference between %d and %d is %d\n", a, b, diff);
      break;
    case '%': //will be used for modulus.
      printf("Enter the first and second number: ");
      scanf("%d %d", &a, &b);
      rem = a%b;
      printf("The remainder of %d and %d is %d\n", a, b, rem);
      break;
    case '*': //will be used for product of 2 no.
      printf("Enter the first and second number: ");
      scanf("%d %d", &a, &b);
      multi = a*b;
      printf("The multiplication of %d and %d is %d\n", a, b, multi);
      break;
    case '/': //will be used for the division.
      printf("Enter the first and second number: ");
      scanf("%d %d", &a, &b);
      div = a/b;
      printf("The division of %d and %d is %f\n", a, b, div);
      break;
    default:
      printf("Error! character please retry");
    }
}

Результаты теста:

$ ./dummy
Choose the character you want to use(+, -, *, /, %): %
Enter the first and second number: 5 2
The remainder of 5 and 2 is 1
$