Jun 09, 2008
RAY: This is from my agricultural series, but it's more than that. It's agricultural and mathematical and familial and historical.
One day last summer, Tommy decided to take a drive into the countryside. After decades of being surrounded by bricks and concrete he had begun to forget what cows and horses looked like.
After a few hours' drive he arrived at a farm stand. Tommy was so excited by the sight of all those fresh fruits and veggies, that he tripped getting out of his MG and knocked over a huge basket of eggs. And of course they all broke!
He asked the farmer how many eggs were in the basket because he obviously wanted to compensate her for them. She said she didn't know exactly.
"But I do know,' she said, "that this morning I decided the eggs would be easier to sell if they weren't in a big basket but if they were in little paper bags.
"So when I took all my eggs and placed two in each bag I had one left over. I didn't like that so I took the eggs and I put three in each bag. And when I did that, I also had one left over.
"When I put four in each bag, I had one left over. When I put five in a bag or six in a bag, I had one left over. And when I put seven in a bag, I had none left over."
So here's the question: what's the smallest number of eggs that Tommy should have to pay for?
One day last summer, Tommy decided to take a drive into the countryside. After decades of being surrounded by bricks and concrete he had begun to forget what cows and horses looked like.
After a few hours' drive he arrived at a farm stand. Tommy was so excited by the sight of all those fresh fruits and veggies, that he tripped getting out of his MG and knocked over a huge basket of eggs. And of course they all broke!
He asked the farmer how many eggs were in the basket because he obviously wanted to compensate her for them. She said she didn't know exactly.
"But I do know,' she said, "that this morning I decided the eggs would be easier to sell if they weren't in a big basket but if they were in little paper bags.
"So when I took all my eggs and placed two in each bag I had one left over. I didn't like that so I took the eggs and I put three in each bag. And when I did that, I also had one left over.
"When I put four in each bag, I had one left over. When I put five in a bag or six in a bag, I had one left over. And when I put seven in a bag, I had none left over."
So here's the question: what's the smallest number of eggs that Tommy should have to pay for?
Answer:
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!
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!
