Once again, these are advanced strategies for students who know their number facts securely, and are ready to move on to more advanced topics.

The numbers 3 and 9 have an unusual, perhaps unique characteristic: for each of their multiples, the sum of the digits is also a multiple of 3 or 9. For example:

Consider the number 78: Since 7 + 8 = 15, which is a multiple of 3, 78 is likewise a multiple of 3. (We could immediately extend this to any other combination of 7 and 8, such as 87, 780, 8070, etc.)

Think about the number 153: Since 1 + 5 + 3 = 9, 153 is a multiple of 9 (153 = 9 x 17). Again, we could find other numbers with those digits, such as 351, 5013, 13 500, etc.

This feature of multiples of 3 and 9 can be used easily to test any whole number for divisibility by 3 or 9.