Friday, July 6, 2012
The Age of Diophantus
Diophantus of Alexandria (c. 200--c. 284 CE) discovered methods for finding integer or rational solutions to certain types of algebraic equations. A riddle from the 5th century purports to express the number of years that Diophantus lived (it isn't known whether the facts in the riddle are accurate):
Diophantus lived one-sixth of his life as a child, then one-twelfth of his life later he grew a beard. After another one-seventh of his life he married, and five years after that he had a son. His son lived only half as long as he did. Four years after his son's death, Diophantus died. How many years did Diophantus live?
Solution: Since the riddle talks about one-twelfth of Diophantus' life and one-seventh of his life, we guess that the number of years he lived is a multiple of both $12$ and $7$, and hence a multiple of $84$. But the only multiple of $84$ reasonable for a human lifespan is $84$, and we see that $84$ years satisfies the conditions of the problem:
The reasonable guess succeeded. We can also solve the problem directly. Let $x$ be the number of years Diophantus lived. Then
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