Patterns for Numbers Divisible by 3

What the hell is this, you might be wondering? Fair question. I’m guessing maybe one or two people will find this interesting, and—full disclosure—I’m one of them.

After some sleepless nights I discovered a mindfulness meditation exercise on the Headspace app. In it, instead of counting sheep or simply counting backwards, starting at 10,000 you count backwards by threes. I tried this a few times and it worked well for me. But sometimes my mind wandered, and I ended up thinking about number patterns divisible by 3. Turns out, there are some pretty interesting ones I’d never really noticed before. Turns out I also wasn't sleeping.

Divisibility Rule for Three. Let’s start with the basics. There's a simple rule: if the sum of a number’s digits is divisible by 3, then that entire number is divisible by 3. Simple enough. For example, 15 => 1 + 5 = 6, which is divisible by 3, so 15 is also divisible by 3. Another example: 4,838,010 => 4 + 8 + 3 + 8 + 0 + 1 + 0 = 24, and taking apart the '24' we get 2 + 4 = 6 which we all know is divisible by 3.

Since every third positive integer is divisible by 3, you can guess there are a bunch of patterns. What are they and how can we stumble on them? Here are a few that surprised me—mostly because I’d never actually stopped to think about them.

• All 3-digit numbers where the digits are the same: 111, 222, 333, ... 999. According to the rule, for 111, for instance, 1 + 1 + 1 = 3 x 1. Boom, divisible by 3. The same logic holds for the rest (4 + 4 + 4 = 3 x 4 = 12, etc.).

• Numbers of any length where the number is repeated a factor of 3 times: Think 111000, 110001, 101001, 100011—you get the idea.

• All digits are repeated a factor of 3 times: 111222333444, 123123123, 945945945. It doesn’t matter what the digits are; repeating them in multiples of 3 does the trick.

• All 3-digit numbers whose digits are consecutive: 345, 456, 678… or any permutation of those consecutive digits, like 534 or 645. Why? In any set of 3 consecutive numbers, one will have a remainder of 1 when divided by 3, another a remainder of 2, and the last will divide evenly, adding up to a total divisible by 3, satisfying the rule.

• Numbers Made from Consecutive Sequences, Repeated. We talked about 3-digit consecutive sequences like 345 or 456. But guess what? If you take that sequence and repeat it, you’re still in multiple-of-3 territory. For example, 456456 and 123123 remain multiples of 3 because each chunk individually sums to a multiple of 3, and repeating a multiple-of-3 chunk keeps the total sum in line with the Divisibility Rule for Three

Anyway, just a fun little something I discovered while trying to cure my insomnia. Not sure if it’ll help you get more sleep, but it might give you a neat party trick (if you attend my kind of parties)—or a bizarre conversation starter—next time you need one.

Happy counting!