The Repdigit Trick: Why 111 Splits by 37 But Not 11

Añadido: 2026-05-06

[00:00:00]Not only is it true that 111, 222, 333, and so on all the way up to 999 do divide by 37, but don't divide by 11.

[00:00:11]It's also true that the number 111 septilian, 222 seextilian, 333 quintilion, 444 quadrillion, 555 trillion, 666 bill, 777,888,999 itself divides by 37. but not 11. And here's a relatively easy way to know.

[00:00:33]We'll start with something simple like 111 itself. So although it doesn't look like it, 111 does divide by 37 and does not divide by 11. 37 goes into 111 three times. But if you tried to divide 111 by 11, it would go in 10 times with a remainder of 1. Because of the nature of multiples, this also means that 222 and 333 and any other multiple of 111 is itself also a multiple of 37. Since 222 is 111 * 2, 22 / 37 is also just 111 / 37 * 2. That is it's 3 * 2 makes 6. But furthermore, this means that we can kind of scrunch these numbers together and the number 111,222 will also divide by 37. If you were doing the long division, you would see that the 37 goes into 111 three times.

[00:01:31]And then you'd bring down the first two.

[00:01:33]37 doesn't go into that. Bring down the second two. 37 doesn't go into 22. But then 37 goes into 222 six times. And so we get the quotient 3006. You can extend this logic such that 37 will go into this absolutely massive number. 3 006 009 012 015 018 021 024 and 027 times. So that three septilian etc is the quotient for 111 septilian etc divided by 37. Now remember when we tried to divide 111 by 11, we had a remainder of 1. And it is possible that those remainders, if we stack them up and divide this larger number by 11, could group together in such a way that eventually gives us a multiple of 11. But that doesn't end up happening here. If you divided this absolutely crazy number by 11, what you would end up with at the end of that process is a remainder of five. If you like this sort of math, I would strongly encourage you to check out the link in bio. As they say, I have partnered with studio.com to create a numbers sense coaching app. That is an app that would help you begin to think through numbers and the amazing connections that are possible between them. So, I hope you will check that