In the previous post, we have discussed about divisibility by 2. In this post, we discuss about divisibility by 3.
Rule: A number is divisible by 3 if the sum of the digits is divisible by 3.
The number 321 is divisible by 3 because 3 + 2 + 1 = 6 is divisible by 3. On the other hand, the number 185 is not divisible by 3 because 1 + 8 + 5 = 14 is not divisible by 3. Now, why does this rule work?
Notice how the numbers are represented in expanded notation:
This means that number in hundreds can be represented as
where h, t, u are the hundreds, tens, and units digits. Now, we can represent as
and regroup the terms as
Of course, is divisible by 3, so it only remains to show that is divisible by 3. But, is the sum of the digits of a 3-digit number. This proves (for three digit numbers) that the rule above is true.
Although the proof above works only for 3-digit numbers, it can be done to any number of digits.