Math geekery fail

I apologize up front for the insane nerdiness of this post.

So the other night I was lying in bed, unable to sleep, and I started to think about numbers. It’s actually not something I do a lot, but every once in awhile, especially when my brain wont shut up long enough for me to go to sleep, I’ll try to bore it to sleep with math. On this particular night, I started thinking about number systems and units of measure.

People are quick to deride the Imperial system of measurement we use here in the US as “archaic” and “illogical.” Most of the time people point to the ease of converting between units in the decimal based metric system as evidence of its superiority. And no doubt about it, the metric system shines if what you want to do is convert between units. However, if what you mainly want is the ability to divide things into equal pieces easily, that’s where the Imperial system has the metric system beat hands down. The entire reason a foot has 12 inches and a yard has 3 feet is because 12 is evenly divisible by 2, 3, 4, and 6. So if you have a piece of cloth 4 yards long and you want to divide it into thirds, the math comes out even–each piece should be 48 inches (4 feet) long. Good luck doing the same with the corresponding metric units. A piece of cloth 4 meters long cut in thirds would have each piece be 133.33~ centimeters long. Have fun finding a ruler that can handle that easily.

Now, this isn’t meant as a defense of the Imperial system. It’s all just my usual roundabout way of getting to the point. As I lay there thinking about this stuff, I started thinking how useful (for a very specific and narrow definition of “useful”) it would be to find a number that would be evenly divisible by all the integers up through 10. The Babylonians used a base 60 system, specifically because it has so many useful small factors (2, 3, 4, 5, 6, 10, and 12.) They’re also the ones who first divided the circle into 360 equal parts for exactly the same reason (the factors of 360 include 2, 3, 4, 5, 6, 8, 9, 10, and 12.) 360 is really very close to what I was looking for.  It lacks only the number 7 as a factor, in fact!

Wow, so close, and yet, so far away! I lay there in my sleep deprived state, and turned the problem over and over in my mind. The only answer I came up with before finally falling asleep was that there was at least one number that was evenly divisible by 2, 3, 4, 5, 6, 7, 8, 9, and 10, and that’s 10 factorial, which is 3,628,800.  The factorial operation means to multiply together all integers that are less than or equal to the number being operated on.  So 10 factorial is the same as 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800.

That answer satisfied me enough to finally let me sleep. However, I kept being bugged by the issue in days to come. 3,628,800 is a pretty big number. It sure seems like there must be a smaller number that works, but I couldn’t for the life of me puzzle out what it could be. I would occasionally think of a number that might work, and whip out a calculator to test it, and inevitably it wouldn’t work out. The problem baffled me.

Until today. Today I realized with a sudden flash of the blindingly obvious that the answer I was looking for was 2520. How did I come up with that number? Simple. 360 x 7 = 2520. I already knew that 360 was divisible by all the other numbers I cared about, so what I was really looking for all this time without even realizing it was a number that was evenly divisible by both 360 and 7. The easiest way to find a common multiple between any two numbers is to simply multiply them together.

Ah, but is this the smallest number that works? I puzzled for a bit longer. It was clear to me that just multiplying two numbers does not automatically give you the smallest number that is evenly divisible by both. 6 x 4 = 24, and yet 12 is also evenly divisible by both 6 and 4. If only there was some way to discover the smallest possible number that is evenly divisible by two or more given numbers. Some kind of… I don’t know, some kind of least common multiple.

…

Yes, ladies and gentlemen. It took me weeks to realize the answer to my conundrum was to apply a little grade school math. Hell, they probably teach this shit on Blue’s Clues now. It’s easy enough to find an LCM with a bit of tedious calculating, but the interwebs makes it trivial. Just go over to Wolfram Alpha and punch in “lcm 2,3,4,5,6,7,8,9,10″ and in like half a second it spits out 2520. A second’s further thought had me slapping my head, since of course the least common multiple of 7 and 360 was simply 7 x 360, because 7 is fucking prime. So the only real question I should have had this whole time is whether 360 is the LCM of 2, 3, 4, 5, 6, 8, 9 and 10. And of course it turns out it is. Those Babylonians knew their shit.

Unlike me.