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:
\[
\frac{84}{6}+\frac{84}{12}+\frac{84}{7}+5+\frac{84}{2}+4=14+7+12+5+42+4=84.
\]
The reasonable guess succeeded.  We can also solve the problem directly.  Let $x$ be the number of years Diophantus lived.  Then
\[
\frac{x}{6}+\frac{x}{12}+\frac{x}{7}+5+\frac{x}{2}+4=x.
\]
Hence
\[
9=x\left(\frac{9}{84}\right),
\]
and $x=84$.

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