Divisibility By Three, Six & Nine

When people say that being able to do simple math is what counts the most, they may not be far off. Did you know that as long as you can add, you can look at any number and tell whether or not the number is evenly divisible by three and/or nine?

Let’s start off with a short example. Do you know if 528 is divisible by 3 and/or 9? To figure it out, you can do the following:

1. Add all of the digits together:
   5 + 2 + 8 =  15
2. Since 15 is still more than one digit long, add up those digits:
   1 + 5 = 6
3. Since the one digit isn’t a 9, we automatically know that 528 isn’t divisible by 9.
4. Since 3 fits into 6 evenly (3 + 3), we know that 528 is evenly divisible by 3.

Believe it or not, this process works for all numbers!  Here is an example of checking a really long number:  1234567890.

1. Add all of the digits together:
   1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 = 45
2. Since 45 is still more than one digit long, add up those digits:
   4 + 5 = 9
3. Since the one digit is 9, we know that 1234567890 is evenly divisible by 9.  (By the way, since it is divisible by nine, it must be divisible by 3.)
4. Since 3 fits into 9 evenly (3 + 3 + 3), we know that 1234567890 is also evenly divisible by 3.

It is important to note that you could stop at step 2 since 45 is evenly divisible by 9.  The only reason I took it a step further was to show the easiest way to determine divisibility for people who aren’t necessarily good at math.

For those of you who are wondering why this three and nine process works, it is due to the fact that we represent numbers in base 10.  If we represented our number in base 8 (octal), we would be able to use this same process to easily determine divisibility by 7.  If we represented our number in base 16 (hexadecimal), we would be able to use this same process to easily determine divisibility by 3, 5, and 15.

What about divisibility by 6? Since we already know how to determine divisibility by 3, all we have to do is also check whether or not the original number is even. If it is divisible by 3 and even (divisible by 2), this means that it is divisible by 6.