Math Fact of the Day - How to easily tell if a large number is evenly divisible by three.

2,389,202,392,398,204 is a very large number, but you can easily tell that this number is evenly divisible by three. You can do this by simply adding the sum of it's digits then checking if the digits sum to a number that is evenly divisible by 3. The trick is that if the sum of a number's digits is evenly divisible by 3, then the original number is also evenly divisible by 3.

Before we jump into explaining the trick, here are some numbers that are divisible by 3:
(When divided by 3, the remainder is 0).
30 (30 / 3 = 10 with a remainder of 0)
18 (18 / 3 = 6 with a remainder of 0)
12 (12 / 3 = 4 with a remainder of 0)

Here are a couple of examples that show numbers that are not divisible by 3:
(When divided by 3, the remainder is 1 or 2)
31 (31 / 3 = 10 with a remainder of 1)
20 (20 / 3 = 6 with a remainder of 2)

Next, here is a couple of examples explaining the trick.

Example 1
First let's try out the trick on a smaller number. Let's choose the number 3,213.
First we add the digits 3+2+1+3 = 9
Now we evaluate the number 9 and since 9 / 3 = 3 with a remainder of 0, we can conclude that 3213 is also divisible by 3 (In fact 3213 / 3 = 1071 with a remainder of 0).

Example 2
Now let's check another number 234,243, which is also divisible by 3. If we wanted to apply our trick to check if 234,243 is divisible by three, we simply add the digits of our number 234,243:
2+3+4+2+4+3 = 18 and 18 is divisible by three (18 / 3 = 6 with a remainder of 0), which implies that our original number 234,243 is also divisible by 3. If we we did not know if 18 was divisible by three we could further simplify by adding the digits of 18, which is 1+8 = 9 and then check to see if 9 is divisible by 3. Since 9 is divisible by 3 (9 / 3 = 3 with a remainder of 0) we can infer that 18 is divisible by 3, and then since 18 is divisible by 3 as well, then our original number 234,243 who's digits summed to 18 is also divisible by 3.

Ok now lets break down the big number 2,389,202,392,398,204
First we will add the digits.
2+3+8+9+2+0+2+3+9+2+3+9+8+2+0+4 = 66
Now let's pretend that we are unsure if 66 is divisible by 3. We could then take this a step further and add the digits of 66.
6+6 = 12

Then, if for some reason we were unsure if 12 was divisible by three we could go one more step and add the digits of 12.
1+2 = 3

At this point we can obviously see that our sum 3 is divisible by 3, (3 / 3 = 1 with a remainder of 0) indicating that 3 is of course divisible by 3.

Then we go back up our list and say, since 3 is divisible by 3 then 12 is also divisible by 3 since 12's digits sum to be 3 (1+2 = 3).

Next, since 12 is divisible by 3, 66 must also be divisible by 3 since 66's digits sum to equal 12 (6+6 = 12)

Finally, since 66 is divisible by 3, we can conclude that 2,389,202,392,398,204 is also divisible by 3 since the sum of 2,389,202,392,398,204's digits equal 66 (2+3+8+9+2+0+2+3+9+2+3+9+8+2+0+4 = 66)

I hope you enjoyed this tutorial, please check back soon for more science, mathematics, and programming tutorials!

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