## Past Political Life

In 1999, while collecting information on an important politician figure, I found that his age at death was one thirty-ninth of the year of his birth. How old was the politician in 1921?
Source: mathcontest.olemiss.edu 2/11/2008

SOLUTION
Let $x$ be the year of birth. Let $y$ be the year of death. Then, age at death equals $y-x$ and we have the following equation:
$y-x=x\div 39$        (1)

We have the following facts:
1. $x$ is a multiple of 39 because $y-x$ is a whole number
2. $x<1921$
3. $y>1921$

Rewrite Eq. (1) and apply a few numbers for age at death $y-x$:
$39\times \left (y-x\right )=x$
$39\times 47=1833$
$39\times 48=1872$
$39\times 49=1911$
$39\times 50=1950$

We eliminate $x=1833$ because $y=1833+47=1880\ngtr 1921$. Likewise, we eliminate $x=1872$ because $y=1872+48=1920\ngtr 1921$.

We also eliminate $x=1950$ because $1950\nless 1921$.

The process of elimination leaves us with the solution:
Year of birth $x=1911<1921$
Year of death $y=1911+49=1960>1921$
Thus, the politician’s age in 1921 is $1921-1911=10$.