Yes the patterns repeat every 15 integers. So you only need to do one division operation, get the index modulo-15.

I was bored once at work and figured out a way to compute this without doing much arithmetic. It only requires 1 modulo operation.

   for (int i=1; i<100;i++)
   {
       // make a bit vector with a 1 in ith bit, modulo 15.
       unsigned int i_as_bitvector = 1 << (i % 15); 
       // Check it against the valid positions for FizzBuzz

       // 0ctal 011111 is binary 1001001001001          
       // Octal  02041 is binary   10000100001
       printf("%d  ", i); 
       if (i_as_bitvector & 011111 ) printf("Fizz"); 
       if (i_as_bitvector &  02041 ) printf("Buzz"); 
       printf("\n");
  }

I also have a version which has NO magic constants anywhere in the program, except 3 and 5. I'll post it if someone is interested.