Special Blog Post: Scholarly Snippets and Quantitative Quandries, Solution Post #1

This is the first in a small series of posts offering solutions and insights into problems posed in my Special Blog Post of December 12, 2006, "Scholarly Snippets and Quantitative Quandries." As I noted in the comment thread for that article, a goodly number of respondents were hitting right on or close to the correct answers and explanations for the problems and curiosities I offered. With a nod to nightshift66 and others who offered explanations for what was going on, below is one of the curiosities, a math trick, that was presented in the post:

Grab your calculator. It doesn't have to be anything fancy, just one that does basic arithmetic operations. Don't forget to turn it on first.

Choose a three-digit number. It can be any number from 100 to 999. Key it in to your calculator.

Now, repeat the digits in the same order so you have a six-digit number displaying. For example, it you had chosen 749 to begin with, you should have 749749 now in the display window.

Remember, I don't know what three-digit number you initially chose, so I certainly don't know the six digit number, right?

But I do know that, whatever number you have on that display, it's divisible by 13. That's right, I know for a fact that it's divisible by 13. Go ahead: hit the "÷" button, then key in "13" and hit "=" to see that I'm right. You've got a whole number displaying, don't you?

But, wait; there's more. I know for a fact that the number you now have displaying on your calculator is divisible by 11. Oh, yes it is. Try it: hit the "÷" button again, then key in "11" and you'll see a whole number show up after you hit the "=" sign. Pretty neat, yes?

And here's the grand finale. The number you now have on your calculator is divisible by 7. Do it: hit the "÷" button one last time, then key in "7" and hit the "=" button.

Lo and behold, not only was I right, but you should notice a much cool something at this point. What is it you see on your calculator's display?

As it turns out, if you follow the instructions, you'll find that, at the end of the run of calculator key strokes, you'll end up with the same three-digit number you started with, no matter what number it was!

So, here's the somewhat formal explanation of why this trick works. I've laid off the algebra as much as possible, but there's still a little, especially with respect to what's called "factoring," which is where the same multiplier of two or more mathematical terms being added or subtracted is pulled out and represented just once outside parentheses, as in 12x+21y being the same as 3(4x+7y).

Suppose we represent a three-digit number by xyz, where x is the hundreds value, y is the tens value, and z is the ones value. So, for example, the number 749 would be such that x=7, y=4, and z=9. Now, we'll write the xyz more formally as

100x+10y+1z.
In the example using 749, we can say that this number is really 100·7+10·4+9·1. In fact, any number can be written in this form of descending powers of ten. Notice that this means any number, in base 10 or any other base, is nothing but a polynomial (a sum of powers of the same base).

Okay, we have xyz now representable as 100x+10y+1z. So, let's do the repeating of the digits to get xyzxyz, as in the example where 749 became 749749. Following the same procedure for re-casting as we did for the three-digit number xyz, we can do xyzxyz as

100000x+10000y+1000z+100x+10y+1z.
In the example with 749, which could be written as 100·7+10·4+1·9, we could write the number 749749 as 100000·7+10000·4+1000·9+100·7+10·4+1·9.

Summarizing where we are, if we start with the number xyz and then repeat the digits, we're going to end up on our calculator with a number that, mathematically speaking, is 100000x+10000y+1000z+100x+10y+1z.

Let's do some re-arranging of terms, here:

100000x+10000y+1000z+100x+10y+1z
from above can be rearranged as follows (using parentheses to group terms we'll want to play with in just a minute):

(100000x+100x)+(10000y+10y)+(1000z+1z)
Okay, now we're going to pull a couple of common factors out of each of those parentheses groups. Notice that the first little group has a common factor of x and a common factor of 100, the second little group has a common factor of y and a common factor of 10, and the third little group has a common factor of z and a common factor of 1 (which is rather trivial, but it's worth noting just for the record). Here we go with pulling the common factors out of each grouping:

100x·(1000+1)+10y·(1000+1)+1z·(1000+1)
Notice at this point that we have (1000+1) three different places, so in all three places we'll do that little bit of arithmetic:

100x·(1001)+10y·(1001)+1z·(1001)
And now we see that we have a three-term expression, where all three terms have the common factor of 1001, so we can factor that off to have the following (putting the common factor of 1001 on the back side of the resulting expression):

(100x+10y+1z)(1001)
Oh, but wait! Look at that first thing, the (100x+10y+1z): that's just the fancy way of writing the number we started with, xyz! Well, spank me hard and call me Florence: the number xyzxyz is nothing but xyz·1001, for Heaven's sake. So a six-digit number like, say, 749749 is nothing but 749·1001. Ditto for any other six digit number of the same form. As long as the first and second pairs of three digits are identical, the six-digit number will be divisible by 1001, and the result will be one of those identical pairs. So, for another example, 358358 is two identical pairs of 358, so 358358 will be divisible by 1001, and the result of the division (what's called the "dividend") will be 358!

The trick I pulled in the problem was to press into service the rather curious fact that 1001 is 7·11·13, so all I was doing by that run of "divide by this, then divide by that, then divide by the other" was a drawn-out version of having you divide the six-digit number by 1001, which I knew would return you from the six-digit number to the three-digit number you started with.

Very cool, yes?

Okay, this gives you an idea of why it's better to know me in cyberspace than in real life: on the Internet, you can close your browser window, and I go out of existence. Before you do that, though, I've got one more totally arcane piece of trivia about this problem that will probably make you not only want to close the browser window, but also want to clear your Internet cache just to make sure I'm really gone.

Remember that number 1001, and how it's 13·11·7? Here's something strange: 1001 is actually not just a number in base 10; it's also a valid number in base 2 (where only the numbers 0 and 1 exist, the basic language of computers). Well, it so happens that the number 1001 in base 2 is the number 9 in base 10.

Yeah, so? Well, so, remember 13·11·7? Did you notice that this is an ordered sequence of odd numbers (in base 10), except that one number is missing from the sequence? Yes, it's the 9, which is 1001 in base 2, and the 9 is missing from the number 2 position from smallest to largest in the sequence.

Yo. Is that cool, or what?

Okay, if you think it's cool, you're more of a math geek than you probably want to admit. We'll do lunch at the QuantHeads Café next week.



The Dark Wraith will post some more stuff tomorrow.