The sum of the reciprocals of three positive integers is $\frac{13}{12}$. What are the three positive integers?

Answer 1

Let the three positive integers be $a$, $b$, and $c$.  We have:

$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{13}{12}$

Since $a$, $b$, and $c$ are positive integers, we can assume that $a \leq b \leq c$.

Let's try different values for $a$ and see if we can find $b$ and $c$ that satisfy the equation.

If $a=1$, then $\frac{1}{b} + \frac{1}{c} = \frac{13}{12} - 1 = \frac{1}{12}$.  The only possible solution in positive integers for this equation is $b=12$ and $c=12$.  However, this is not a valid solution since $a$, $b$, and $c$ should be distinct.

If $a=2$, then $\frac{1}{b} + \frac{1}{c} = \frac{13}{12} - \frac{1}{2} = \frac{7}{12}$.  The only possible solution in positive integers for this equation is $b=3$ and $c=4$.  So one solution is $a=2$, $b=3$, and $c=4$.

Therefore, the three positive integers are $2$, $3$, and $4$.