Tutorial Exercises: Set 4

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Exercise 1

Write a function that takes two positive integer arguments a,b and returns the floating-point value f(a,b) = (a²+b²)/(ab-1).

Write a main() function that reads a bound B from the user and computes f(a,b) for 1 <= a < b <= B. If f(a,b) happens to be an integer, the main() function prints the value of that integer along with the corresponding values of a and b.

Solution 1

   #include <stdio.h>
   #include <stdlib.h>

   double f ( unsigned long int a , unsigned long int b )
   {
      if (a*b <= 1) return 0;
      return ((double)(a*a + b*b) / (double)(a*b - 1));
   }

   int main ()
   {
      unsigned long int a, b, B, t;
      double fab;

      printf("Enter a bound: "); scanf("%lu", &B);
      for (a = 1; a <= B; ++a) {
         for (b = a+1; b <= B; ++b) {
            fab = f(a,b);
            t = (unsigned long int)fab;
            if (fab == t) printf("f(%lu,%lu) = %lu\n", a, b, t);
         }
      }
      exit(0);
   }

Exercise 2

Write a function that computes the gcd of two integers.

Write another function that uses the above function for computing the lcm of two integers.

Solution 2

   int gcd ( int a , int b )
   {
      int r;

      if ((a == 0) && (b == 0)) return 0;
      if (a < 0) a = -a;
      if (b < 0) b = -b;
      while (b > 0) {
         r = a % b;
         a = b;
         b = r;
      }
      return a;
   }

   int lcm ( int a , int b )
   {
      int l;

      if ((a == 0) && (b == 0)) return 0;
      l = a * b / gcd(a,b);
      if (l < 0) l = -l;
      return l;
   }

Exercise 3

Write a function that accepts two floating point numbers x,y and a positive integer n as input. The function prints all the n-th roots of the complex number x+iy.

Solution 3

   void printRoots ( double x , double y , unsigned int n )
   {
      double r, t, s;
      unsigned int i;

      if (n == 0) return;
      if ((x == 0) && (y == 0)) r = t = 0;
      else {
         r = pow(x*x + y*y, 1.0/(double)(2*n));
         t = atan2(y,x) / (double)n;
      }
      s = 2 * 3.1415926535 / n;
      for (i = 0; i < n; ++i) {
         printf("%d-th root: (%lf) + i(%lf)\n", i, r*cos(t+i*s), r*sin(t+i*s));
      }
   }

Exercise 4

Express the return value of the following function in terms of its input arguments.

   unsigned int f ( unsigned int x , unsigned int y )
   {
      unsigned int d, s;

      s = 0;
      while (y != 0) {
         d = y % 5;
         y /= 5;
         s += x * d;
         x *= 5;
      }
      return s;
   }

Solution 4

xy (The function multiplies x and y in their base-5 representations).

Exercise 5

What does the following program print?

   #include <stdio.h>
   #include <stdlib.h>

   int f ( int a , int b )
   {
      a = a + b;
      b = a - b;
      a = a - b;
      printf("Inside function : a = %d, b = %d\n", a, b);
      return a;
   }

   int main ()
   {
      int a, b, c;

      a = 56; b = 31;
      printf("Inside main()   : a = %d, b = %d\n", a, b);
      c = f(a,b);
      printf("Inside main()   : a = %d, b = %d, c = %d\n", a, b, c);
      a = f(a,b);
      printf("Inside main()   : a = %d, b = %d\n", a, b);
      exit(0);
   }

Solution 5

   Inside main()   : a = 56, b = 31
   Inside function : a = 31, b = 56
   Inside main()   : a = 56, b = 31, c = 31
   Inside function : a = 31, b = 56
   Inside main()   : a = 31, b = 31

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