Divisibility Rules for 3 and 9

First, we list some numbers that are divisible by 3, and some that are divisible by 9. Then we notice that the same is true for the sum of their digits. For divisibility by 9, we can represent a number as the sum of two parts, where one part is divisible by 9, and the other part is the sum of its digits. Therefore, a number is divisible by 9 if the sum of its digits is divisible by 9. In the same way we can show for divisibility rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.

Image credit (Thumbs Up):

[matiaspereira]

3 and 9: The sum of the digits is a multiple of 3 or 9

27 and 81: The difference between 8 times the last digit and the rest of the number is 0 or a multiple of 27 or 81

243 and 729: The sum of 73 times the last digit and the rest of the number is a multiple of 243 or 729

2, 187 and 6, 561: The difference between 656 times the last digit and the rest of the number is 0 or a multiple of 2, 187 or 6, 571

19, 683 and 59, 049: The sum of 5, 905 times the last digit and the rest of the number is a multiple of 19, 683 or 59, 049