The Magnificence of 3, 6 and 9

“I would love to hear your thoughts on the relationship between 3, 6 and 9!” – woahandgo

Thanks to woahandgo for suggesting this intriguing idea!

Let’s address relationship number one that everyone in the room is thinking – 3, 6 and 9 are all multiples of 3. Sure, you can check that 6 ÷ 3 = 2 and 9 ÷ 3 = 3, but with larger numbers it takes time and effort (two things that I hate devoting) to manually divide numbers (particularly if you’re not in grasp of a calculator). Thankfully there is a handy way of checking divisibility by 3 that you may have learnt when you were at school.

The trick is to sum up the all the digits in the number repeatedly, and check if this smaller number is divisible by 3:

81? 8 + 1 = 9 ✓

489? 4 + 8 + 9 = 21; 2 + 1 = 3 ✓

69432? 6 + 9 + 4 + 3 + 2 = 24; 2 + 4 = 6 ✓

Why does this work? You can derive a proof using some basic algebra.

Any integer n can be expressed as:

Divisibility by 3 - 1

a, b, c and d are the digits of the number. We can cleverly manipulate this equation to become:

Divisibility by 3 - 2

When we divide n by three:

Divisibility by 3 - 3

We know that (3b + 33c + 333d + …) must be an integer, so whether n is divisible by 3 or not lies in whether the first part is an integer or not. If the sum of the digits (a + b + c + d +…) is divisible by 3, then the first part is an integer, and thus n is an integer.

Okay, that’s your daily dose of algebra for today.

I came across a much more curious yet obscure relationship number two, which is established in this quote by Serbian-American physicist and engineer Nikola Tesla:

“If you only knew the magnificence of the 3, 6 and 9, then you would have a key to the universe.”

The authenticity of the quote is disputable, but it’s fascinating to know what he meant if it were confirmed to be true.

The short answer is: nobody knows. It has been widely speculated that Tesla was obsessed with the number 3 (as he suffered from OCD). Some of the peculiar things he would do would be to walk around a block three times before entering a building, or always stay in a hotel room whose number was a multiple of three.

You can’t blame him to be honest. He just developed an unhealthy attachment to a number. I guess it happens if you do too much maths.



5 thoughts on “The Magnificence of 3, 6 and 9

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