So, I know this was brought up in class, but I have realized more and more over the last few years (especially since we are NEVER allowed to use calculators in any of our math classes) how much I rely on the "divisibility rules" that we all probably memorized in elementary school when we were learning how to divide big numbers (as in two or three digit ones) by single digit numbers for the first time. The divisibility rule about the sum of the digits being divisible by 3 was brought up in class, and we did a few proofs showing divisibility by 5 and one by 17 in class, but I started to wonder why exactly these divisibility rules always work! We always just accept them to be true, and some are really obvious… like divisibility by 2 is obvious because all even numbers are divisible by 2, and divisibility by 6 even makes a ton of sense to think about logically because if something is divisible by 2 and 3 (which is the rule for divisibility by 6), then it has to also be divisible by 6 because 6 is the smallest number divisible by both 2 and 3. But then what about 3? and 9? Why does the sum of the digits matter? Well, it turns out, these examples can be really easily seen by writing out the divisibility in the same way that we do in our proofs. I found this example when I googled divisibility rules and it all makes a ton of sense when you just multiply the digits by their "spot" (as in 100 for the 100's place, 10 for the 10's place, etc). So, why is 567 divisible by 9? Well, most of us would just say because the sum of 5, 6, and 7 is divisible by 9…. but why does that make sense? Here it is!

567 = 5 × 100 + 6 × 10 + 7 = 5 × (99 + 1) + 6 × (9 + 1) + 7 = 5 × 99 + 6 × 9

+5 + 6 + 7

I love it when you see the reasons behind why something you've just accepted mathematically is true.