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

Puzzles and Curious Problems by Henry Ernest Dudeney: $121$

Can you find $2$ numbers composed only of ones which give the same result by addition and multiplication?

Of course $1$ and $11$ are very near, but they will not quite do,

because added they make $12$, and multiplied they make only $11$.

$11$ and $1 \cdotp 1$

Dudeney 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:

$11 + 1 \cdotp 1 = 1 \cdotp 21 = 11 \times 1 \cdotp 1$

$\blacksquare$