Alice said to the twins, “Instead of passing the day rattling each other's bones, why don't you get some gainful employment? I have just imported some apples and bananas into Wonderland. Each kind of fruit comes in boxes each of which holds exactly the same number, apples or bananas. I do not remember how many there are in each box, but it is more than 1. I do know that there are 289 apples and 221 bananas in the boxes.”

“We will give it a try,” said Tweedledum doubtfully. “I will sell apples and my twin brother will sell bananas.”

“We need to know how many fruit there are in each box,” said Tweedledee.

“Well, see if you can figure that out before I come back,” said Alice. “Perhaps until you do, you should just sell whole boxes of fruit.”

“That is a good idea,” said the twins together.

The Duchess's Cook came in shortly after Alice took off, went to Tweedledum and said, “I see that you have more boxes of apples than Tweedledee has boxes of bananas. I want to buy as many boxes of apples as he has boxes of bananas.”

Tweedledum made the sale, and recorded that now he had 289 − 221 = 68 apples left.

The next three to come were the Dormouse, the March Hare and the Mad Hatter. Each of them bought as many boxes of bananas from Tweedledee as Tweedledum had boxes of apples. Tweedledee recorded that now he had 221 − 68 − 68 − 68 = 17 bananas left.

The next four customers were the Two, the Five, the Seven and the Knave of Hearts. Each bought as many boxes of apples from Tweedledum as Tweedledee had boxes of bananas. Now Tweedledum had no apples left.

Tweedledee said, “I still have 17 bananas left.”

“They are still inside unopened boxes because neither of us had opened one at any time. This means that the number of bananas in each box is a divisor of 17.”

“The only divisors of 17 are 1 and 17,” said Tweedledee. “Since Alice told us the number of bananas in each box is not 1, it must be 17.”

“Let me check,” said Tweedledum. “We have 221 ÷ 17 = 13 and we have 289 ÷ 17 = 17.

So 17 indeed divides both 221 and 289.”