Saturday, February 2, 2008

'Left-Overs' before the Super Bowl: Crazy Eights, Squares, Remainders and Algebra

Ok, so most normal people are not thinking about the significance of the digit '8' in 2008 the day before the Super Bowl. Sorry, but in this post there will be no predictions about the score, no 'over-unders', no boxes, no betting at all. You do have to admit that this is a great time for lovers of mathematics. People are actually interested in mathematical odds and chances of all kinds of weird number combinations occurring in the score on Sunday night. However, this post will focus instead on the number 8, the units' digit in 2008. The Super Bowl comments above will no doubt soon become outdated but the mathematics below will live on! Who knows, maybe the number 8 will turn out to have special significance on Feb 3, 2008? Remember, I said that here before the game!!

2008 is a special number for so many reasons, being divisible by 4 of course: Leap Year, Prez Election year, Summer Olympics and much more. In fact, 2008 is divisible not only by 4 but also by 8 itself. In the good ol' days, some students were even taught the divisibility rules for 2, 4 and 8:
Divisible by 2: If the 'last' digit is divisible by 2 (of course!)
Divisible by 4: If the number formed by the last TWO digits is divisible by 4
Divisible by 8: If the number formed by the last three digits is divisible by 8.

Let's demonstrate this for 2008:
2008 us divisible by 2 because 8 is divisible by 2
2008 is divisible by 4 because '08' is divisible by 4
2008 is divisible by 8 because '008' is divisible by 8

A little weird with those zeros and not particularly interesting, right? Anyone care to guess a rule for divisibility by 16? Interesting, but none of this is the issue for today....

Today, we are are interested in the squares of numbers and their remainders when divided by 8. Notice that 42 is divisible by 8 but 62 is not. So we cannot say that the square of any even number is divisible by 8. What about the squares of odd numbers when divided by 8?
12 leaves a remainder of 1 when divided by 8
32 leaves a remainder of 1 when divided by 8
52 leaves a remainder of 1 when divided by 8
72 leaves a remainder of 1 when divided by 8

What is going on here? That's for your crack investigative team to decipher.

TARGET AUDIENCE: Our readers of course; Middle schoolers through algebra

1. Discover, state and prove a general rule for the remainder when the square of an even number is divided by 8.
2. Discover, state and prove a general rule for the remainder when the square of an odd number is divided by 8.

(1) These are well-known relationships and not very difficult questions. Just something to extend thinking about divisibility, remainders and the use of algebra to deduce and prove generalizations. Prealgebra students may be able to explain their findings without algebra!
(2) 'Discovering' or stating the rule for question (2) is transparent from the examples above. Instructors may prefer 'data-gathering' and making a table first. That is, have students develop a table for the squares of the first 10 positive integers and their remainders when divided by 8. Proving the result for the squares of odd integers is more challenging, even algebraically. Most will see the remainder when dividing by 4, but 8 is slightly trickier.
(3) Those who are more comfortable with congruences and modular arithmetic can approach these questions another way.


Andrée said...

thank you for these. I always need ideas like this.

Anonymous said...

I can't draw this here, but there is nice analytic work kids can do on an 8x8 mat. 8x8 and 4x4 squares are groups of 8, and then the edge is the leftover or the the shortage on the edge can be seen, counted.

Jumping from 4 odd numbers to all odd numbers is big, but not that big.


Eric Jablow said...


And then there are those problems that, though not trivial, teach mathematical reasoning. For example, take a chessboard, and remove squares a1 and h8. Can the remainder of the board be covered by non-overlapping 1×2 dominoes that don't extend past the board?

mathmom said...

Erik, I love that one. I have a whole list of "parity" related problems. Of course the problem is that if I present them as "parity problems" it gives away the answer, or at least most of the answer....