1. Think of two numbers
      (1) the number of the month in which you were born and
      (2) the number of the day of the month on which you were born.

2. Multiply the month number by 5.

3. Add 6.

5. Add 9.

6. Multiply by 5.

7. Add the Day on which you were born.

8. Now get the final answer from your friend.
      Subtract 165 to reveal the month and day that your friend was born.
      The last two digits give the day,
      and the other digits give the month.

Why don't you use some algebra to prove it?
Then following the instructions above:
1. Let m = month and d = day.
2. Multiply the month number by 5:       5m
3. Add 6:       5m + 6
4. Multiply by 4:       20m + 24
5. Add 9:       20m + 33
6. Multiply by 5:       100m + 165
7. Add the Day on which you were born:       100m + d + 165
8. Subtract 165:       100m + d. (The 100 just moves the month over 2 places)