# First function:

def independent(list_a, list_b):
   '''
      We can say that two lists are independent if no value occupies the
      same position in both lists.  Independent lists are not
      required to have the same length.

         For example, [1,2] and [2,1,3] are independent, and
         [1,2,3]  and [4,2,'a'] are not independent.
         
      Parameters:
         list_a and list_b must be lists

      Returns True if list_a and list_b are independent, and
      returns False if they are not
   '''
   if type(list_a) is not list  or type(list_b) is not list:
      return False
   else:
      min_length = min(len(list_a), len(list_b))
      for i in range(min_length):
         if list_a[i] == list_b[i]:
            return False
      return True
      
      
# Second function:   

def primes_up_to_n(n):
   '''parameter :
          n - integer
          
   returns a list of all primes <= n
   returns None if n is not an integer
   '''
   if type(n) is not int:
      return None
   else:
      list_of_primes = []
      for potential_prime in range(2,n+1):
         #print("potential prime",potential_prime)
         could_be_prime = True
         done = False
         possible_divisor = 2
         while could_be_prime and possible_divisor < potential_prime:
            #print("\tpossible divisor",possible_divisor)
            if potential_prime % possible_divisor == 0:
               could_be_prime = False
               is_prime = False
            else:
               possible_divisor += 1
               
         if possible_divisor == potential_prime:
            list_of_primes.append(potential_prime)
            
      return list_of_primes
      
# Third function:

def binary_string(n):
   '''parameter:
         n - integer
         
      returns a string of "0"s and "1"s giving the binary representation of n
      returns None if n is not an integer
   '''
   
   if type(n) is not int:
      return None
   else:
      if n == 0:
         return '0'
      else:
         negative = (n < 0)
         n = abs(n)
         bin_string = ''
         while n != 0:
            if n % 2 == 1:
               bin_string = '1' + bin_string
            else:
               bin_string = '0' + bin_string
            n = n // 2
         if negative:
            bin_string = '-' + bin_string
         return  bin_string