Question

One day last summer, Tommy decided to take a drive into the countryside.

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 1

9514 1404 393

Answer:

301

Explanation:

In order for the remainder to be 1 when the number is divided by 2, 3, 4, 5, or 6, it must be 1 more than a common multiple of those numbers. The least common multiple is ...

3·4·5 = 60

So, we want a number that is 1 more than a multiple of 60 and is divisible by 7. This gives us the Diophantine equation ...

7m -60n = 1 . . . . . where m and n are integers

Perhaps the easiest way to solve this is to try values of n between 0 and 6. We find that n=5 and m=43 will be the smallest positive value of m that satisfies this equation. This makes the number of eggs be 7×43 = 301.

Tommy should have to pay for 301 eggs.