On my drive home today, at 7:18, I looked at the clock and thought to myself that if you combine the 7 and 1, you get two 8s, which is pretty nice. But then if you combine those, you get 16 and 1+6=7, a different kind of lucky number. Then I wondered what about 8+8+8? With 24, 2+4=6… hey, I see a pattern here! And so I went on in my head as I slowly made my way through traffic, adding up multiples of 8, then 7, then 6 and 5 and 4 and 3… on and on until I found all the patterns!
Some we are commonly taught – when you add together the digits of any number and it can be divisible by 3, the number can be divisible by 3. (You did learn that, right? For example, 105 can be divisible by 3 because 1+0+5=6 and six can be evenly divided by three.) Another is that multiples of 9 add up to 9: for example, 9X7=63 and 6+3=9. You could even take it up a notch and do, say, 9X23=207 and 2+0+7=9. And of course, we all know that multiples of 5 only end in 5 or 0 and multiples of 10 only end in 0.
Now check out these patterns and keep in mind that only 0 adds up to 0, so we’re considering 1-9 only:
digits of multiples of 1 add up to 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, etc. (up one increment each time)
digits of multiples of 2 add up to 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, etc. (up two increments each time 123456789123456789123456)
digits of multiples of 3 add up to 3, 6, 9, 3, 6, 9, etc. (up three increments each time 123456789123456789)
digits of multiples of 4 add up to 4, 8, 3, 7, 2, 6, 1, 5, 9, 4, 8, etc. (up four increments each time 123456789123456789123456789)
digits of multiples of 5 add up to 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, etc. (up five increments each time, or down four increments each time 123456789123456789123456789)
digits of multiples of 6 add up to 6, 3, 9, 6, 3, 9, etc. (up six increments each time, otherwise known as down three increments each time 6543219876543219)
digits of multiples of 7 add up to 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, etc. (up seven increments each time, otherwise known as down two increments each time 7654321987654321987654321)
digits of multiples of 8 add up to 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, etc. (up eight increments each time, otherwise known as down one increment each time)
digits of multiples of 9 add up to 9, 9, 9, etc. (up nine increments each time, otherwise known as 9 the entire time)
digits of multiples of 10 add up to 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, etc. (up one increment each time)
And then the pattern repeats with 11 mimicking the pattern of 2, 12 mimicking the pattern of 3, etc. Am I the only one who finds this cool? That’s the beauty of mathematics – it’s a precise language with very predictable patterns.
