Wednesday, March 31, 2010

Wednesday Math, Vol. 114: Divisibility

What math can I as a teacher expect everyone to know? Very little, actually. The basics of divisibility is a topic the vast majority of people have internalized, even those who tell me they hate math. By the basics, I mean divisibility by 2, divisibility by 5 and divisibility by 10.

Nearly no one is confused by even and odd. Look at the digit in the one's place, the far right side of the number. 0, 2, 4, 6 and 8 are the even digits, 1, 3, 5, 7 and 9 are the odd digits. What it means is that an even number is divisible by 2 and an odd number isn't. Any even number is 2 times some other whole number, like 296 is 2 times 148, and 148 is 2 times 74 and 74 is 2 times 37, but 37 is odd, so it isn't a multiple of 2.

Divisibility by 5 also has a simple rule. The one's place digit must be 5 or 0 for a number to be a multiple of 5. Multiples of 10 have a last digit of 0, which is the both even and divisible by 5, since 10 is 2 times 5. 745 is 5 times 129, but 129 isn't a multiple of 5.

When we get to the rule for divisibility by 3, some students know it and some don't. It's not quite as easy as the rules for 2 and 5, but it's something people can do in their heads. Take the sum of all the digits in the number. If that number is a multiple of 3, the original number is a multiple of 3 and if it's not, the original number was not. Take 7,321,058 for example, as a number I typed on a whim. 7+3+2+1+0+5+8 = 26, and 26 is not divisible by 3, so neither is 7,321,058. On the other hand 7,321,059 has a digit sum of 7+3+2+1+0+5+9 = 27, and that is divisible by 3, so the larger number is a multiple of 3 as well. This rule also works for divisibility by 9, take the digit sum and check if that is divisible by 9. It was called "casting out nines" back in the day, and it was one of the topics taught in Robinson's New Higher Arithmetic back at the end of the 19th Century and in the early 20th Century.

Pictured here on the left is what we call a lattice in math. You can think of this as a three dimensional structure, if you like. There are lines that go left, lines that go right, and a third set of lines that go obliquely to the right. If two circles are connected by a line segment slanting to the right, the higher one is 3 times the lower one. If the connection is left slanting line, the larger number is 5 times the smaller. The lines that go obliquely to the right represent multiplication by 2. Only numbers that are multiples of just 2, 3 and 5 are in this picture, and the largest is 400.

If we included numbers divisible by 7, for instance, we'd need to add lines that were at some new angle like 90 degrees or 120 degrees. That would make this a four dimensional lattice, and when we get past three dimensions, more than a few students get confused. If we wanted to represent all numbers in such a lattice, every prime number would need its own angle, and since there are infinitely many primes, we would get an infinite dimensional lattice.

If you understood everything above the picture, but got a little dizzy reading the explanation of a lattice, don't worry. That's how I designed the lesson. Divisibility by 2 and 5 really is easy, and 3 isn't much harder. The structure of divisibility is beautiful, but it's definitely not something that "everyone should know".

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