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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 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?

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 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 $x=84$.