Clearly, 639 is not divisible by 7. Consider 2079. Sum of its digits = (2 + 0 + 7 + 9) = 18. So, it is divisible by both 3 and 9. Also, (79 – 2) = 77, which is divisible by 7. So, 2079 is divisible by 7. Also, (9 + 0) – (7 + 2) = 0. So, 2079 is divisible by 11. Hence, 2079 is divisible by each one of 3, 7, 9 and 11.
Explanation as extracted from the printed page; notation may be imperfect.