Henry Ernest Dudeney/Puzzles and Curious Problems/121 - Find the Numbers/Solution

by : $121$

 * Find the Numbers

Solution

 * $11$ and $1 \cdotp 1$

Proof
has already raised this question in his Modern Puzzles: $93$ - Sum Equals Product, where he shows that:
 * $y = \dfrac x {x - 1}$

for any pair $x$ and $y$ such that $x y = x + y$.

The only time $x$ and $y$ are both integers is when $x = y = 2$.

So there's a trick to look out for.

Suppose $x = 11$.

We have:
 * $y = \dfrac {11} {11 - 1} = \dfrac {11} {10}$

and we notice that:
 * $\dfrac {11} {10} = 1 \cdotp 1$

We note that:
 * $11 + 1 \cdotp 1 = 1 \cdotp 21 = 11 \times 1 \cdotp 1$