Tommy's Eggy Escapade!

RAY: Here's the answer. This is a classic example of something called the Chinese remainder theorem. And to solve it you can use that and something called modulo arithmetic.

TOM: Of course! I do that all the time.

RAY: But I decided I would never be able to explain it clearly. So I decided I'm going to do something different, a little different explanation. And I'm going to do something I've never done before. I'm going to give the answer right now.

TOM: Go ahead.

RAY: You had to pay for 301 eggs.

TOM: Get out. I'm not paying for any 300 eggs! Forget it.

RAY: You knocked them over. You broke them. And you still have egg in your beard!

TOM: Now this is interesting...

RAY: Well, let's see how I got the 301 eggs. First of all, we know that our number is not an even number. Right? It can't be because if it's two to a bag, we'd have no remainder. When we put five to a bag we also have one left over. So we know that the number of eggs had to end in one.

TOM: Well, how do we know that?

RAY: Well, because all the multiples of five are five, ten, 15, 20 and end in either zero or five. So our number must be one greater than that. We also know that that number 301 has got to be a multiple of seven.

So if we look at multiples of seven that end in one, here's what we get. We get 21, 91, 161, which happens to be seven times 23. The next one that ends in one is seven times 33, which is 231. Do you see a trend here?

TOM: Yeah.

RAY: You may have noticed that in every case these products are obtained by multiplying seven times some number whose units digit is 3. Like 3, 13, 23, 33.

TOM: Yeah.

RAY: OK. But all of those fail. The only one that works is 7 times 43. And 7 times 43 equals 301.

TOM: Of course.

RAY: And that's the answer. Do we have a winner?

TOM: Wow. Yeah. The winner this week is Linda Corey from Roanoke, Virginia. And for having her answer selected at random from the two answers that we got. No, no. From the many, many correct answers that we got, Linda gets a gift certificate for 26-bucks to the Shameless Commerce Division at cartalk.com, with which she can get a brand-new, four-CD set, Field Guide to the North American Wacko, filled with four of our favorite shows.

RAY: And what, pray tell, makes these our favorites?

TOM: We charted the hate mail. And all four of these shows were in the low hundreds.

RAY: Excellent. Congratulations, Linda!

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