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The Age of Diophantus
Diophantus of Alexandria (c. 210  c 290 CE) discovered methods for finding integer or rational solutions
to certain types of algebraic equations. A riddle from a work published around 500 purports to express the number of years that Diophantus
lived (it isn't known whether the facts in the riddle are accurate):

Diophantus lived onesixth of his life as a child, then onetwelfth of his life later he grew a beard. After another oneseventh
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 onetwelfth of Diophantus' life and oneseventh 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 succeeds. We can also solve the problem with algebra. 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 $x=84$.