Math of Ages

Dec 12, 1998

RAY: We're back, you're listening to Car Talk with us, Click and Clack, the Tappet Brothers, and we're here to discuss cars, car repair, and the new puzzler.

TOM: Yes. I'm looking forward to it, I might say.

RAY: I have at my disposal a --

TOM: In your disposal?

RAY: No, but I have at my disposal a plethora of potentially putrid puzzlers, I'm going to use, and I've been tempted to use a few automotive ones because I know people really detest the automotive ones, and that cuts way down on the mail. But I'm going to use this one which I have stolen, well, that's such an ugly word, I've usurped from Martin Gardner with a little twist, I've added a twist to it.

TOM: Interesting.

RAY: Here it is. This is for all the kiddies out there, and for their parents who like to help them with their homework.

TOM: Was this from one of Martin Gardner's books, or was it from Scientific American --

RAY: No, it was written on his underwear! Yes, it was from one of his many, many books that he wrote before he died. He'd have to write any book before he died, because you can't write any, what? After you die! Even if he's still alive, he'd have to write them before he died, wouldn't he?

TOM: That's correct.

RAY: There you go.

TOM: I think we're pretty safe in saying that he wrote this book before he died.

RAY: Get your pencil.

TOM: I've got it. Pencil.

RAY: Pick out a three digit number, any three digit number like --

TOM: I just did!

RAY: 7-8-9, 2-7 --

TOM: I can understand a three digit number concept!

RAY: Now, repeat those three digits. So, if you picked 2-7-1, you got 2-7-1, 2-7-1.

TOM: Yeah, so I write it next to the three digit number?

RAY: Yeah.

TOM: I got it.

RAY: 2-7-1, 2-7-1.

TOM: How'd you know it was 2-7-1, by the way? OK, I got it.

RAY: It could be any three digit number.

TOM: It is a three digit number.

RAY: All right, now, I want you to divide that number.

TOM: This big number?

RAY: By seven.

TOM: I've got it.

RAY: OK, all right, OK, that's good. Now, you have that quotient, if you have a remainder, put it off to the side. Like if the remainder's one, two, whatever it is, put it off to the side. Take the quotient that you got and divide that by 11.

TOM: Eleven? Oh god.

RAY: Another prime number. And whatever remainder you have, put that off to the side too.

TOM: Got it, yeah.

RAY: Now, divide that remaining quotient, the quotient you just got --

TOM: By 17!

RAY: -- by 13!

TOM: Thirteen, I see.

RAY: Another prime number! OK, now take that number --

TOM: Wait, I'm not finished yet!

RAY: Come on, what is this? Third grade!

TOM: Yeah!

RAY: OK, now take that number that you've got, and the quotients, the remainders that you had from the previous divisions?

TOM: Yeah.

RAY: Add them all together.

TOM: Add what together?

RAY: Add the remainders and the last quotient that you got.

TOM: The last one?

RAY: Yeah. The last quotient and the remainders.

TOM: Got it.

RAY: And you're going to wind up with the original number.

TOM: What do you mean the original number?

RAY: The 2-7-1 that you started with!

TOM: I did not --

RAY: You screwed it up!

TOM: Oh no. I did! I ended up with the original number! I did end up with the original number. Which was, I'll tell you now, now that it's no secret anymore --

RAY: 2-7-1!

TOM: Of course, 2-4-7.

RAY: 2-4-7, very good.

TOM: Which had no remainders because everything was evening divisible.

RAY: Sonya Henne's tutu!

TOM: Imagine that.

RAY: So, the question is, why does this work?

TOM: Wow! If you pick any three numbers --

RAY: Any three digit number.

TOM: Repeat it.

RAY: Repeat those three digits. So, you had 2-4-7 --

TOM: So I got 2-4-7, 2-4-7.

RAY: Divide by seven.

TOM: Divide by 11.

RAY: And then put whatever remainder there is, if there is one, you put it aside. Divide by 11, and if there's a remainder, put that aside, and by 13, if there's a remainder, you put that aside. Add up the remainders, and add them to the last quotient that you got, and you're going to wind up with your original number. I want to know, very simply, why does this work?



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?


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.

Get the Car Talk Newsletter