RAY: Hi, we're back, listening to Car Talk, with us, Click and Clack, the Tappet Brothers. And we're here to talk about cars, car repair and, of course, the answer to last week's very interesting Puzzler.
TOM: Martin Gardner, division and magic numbers.
RAY: Oh, this requires a pencil. Get your pencil out.
TOM: I've got one.
RAY: OK. I think last week I asked you to pick out a three digit number.
TOM: Oh, I remember this. Yes, I did.
RAY: What's your number?
TOM: 1, 2, 3.
RAY: It's very unimaginative.
TOM: 1, 3, 2.
RAY: OK, 1, 2, 3 is your number. 1, 2, 3 it is. And I asked you to repeat those same digits, so you have the number --
TOM: Oh, yeah, I have the number 1, 2, 3, 1, 2, 3.
RAY: 123,123. Then I asked you to divide that number by 7.
TOM: Right.
RAY: All right?
TOM: Which I did.
RAY: You did that. And I said, also, if you have a remainder, put it to one side.
TOM: And you lied about that, because I had no remainder when I picked the number last week and I got no remainder when I pick a number this week.
RAY: Different number?
TOM: Different number, completely different number.
RAY: You have no remainder, no matter what. So I did lie about that. So then I asked to divide the number by 11.
TOM: Yeah.
RAY: And, again, you'll get a quotient.
TOM: And you said if there's anything left over, put it --
RAY: Put it to one side.
TOM: To one side. There was nothing left over.
RAY: And then I said, divide that now, what remains, by 13. And if you have a remainder, put that to one side.
TOM: Um hum.
RAY: OK? Take this number that you wind up with, this final quotient, and add it to the three remainders that you had, whatever they were.
TOM: Which were all zeros.
RAY: All zero. And you're going to wind up with the original number. And I asked why that happens.
TOM: [MOAN]
RAY: And the reason it happens is that when you take a number like 1, 2, 3, or 4, 5, 6, or 7, 2, 1, and multiply it by a thousand and one, you wind up with the same number repeated. So if you start with 4, 5, 6, and multiply that by 1,001, you get 4, 5, 6, 4, 5, 6, don't you?
TOM: [MOAN]
RAY: And then all you're doing now is dividing it by the factors of 1,001. Which happen to be, some of which happen to be, 7, 11, and 13. So I knew you weren't going to come out with any remainders.
TOM: You little devil, you.
RAY: And that's the reason it works, no matter what the three numbers are. Do we have a winner?
TOM: Wow, geez. I can see it. Mathematics professors all over the country, giving their little students this problem. Boy oh boy oh boy. And have I got a rant and rave about mathematics. I'll discuss it with you later. But, yes, we do have a winner. The winner is Jim Hanlon from where, Anchorage, Alaska? Wow.
[ Car Talk Puzzler ]
