- #1
Math100
- 807
- 226
- Homework Statement
- (Ancient Chinese Problem.) A band of
pirates stole a sack of gold coins. When they tried to divide the fortune into equal portions, coins remained. In the ensuing brawl over who should get the extra coins, one pirate was killed. The wealth was redistributed, but this time an equal division left coins. Again an argument developed in which another pirate was killed. But now the total fortune was evenly distributed among the survivors. What was the least number of coins that could have been stolen?
- Relevant Equations
- None.
Let be the least number of coins that could have been stolen.
Then , and .
Applying the Chinese Remainder Theorem produces:
.
This means and .
Observe that
Now we have and .
Thus .
Therefore, the least number of coins that could have been stolen is .
Then
Applying the Chinese Remainder Theorem produces:
This means
Observe that
Now we have
Thus
Therefore, the least number of coins that could have been stolen is