## 174. Sum Equals Product.

- Just think about it,- said one man to me,- there are two numbers whose sum equals their product; that is, you get the same result whether you add them or multiply them together. These are 2 and 2, since their sum and product are both 4.

He then made a gross mistake by saying:

- I discovered that these are the only two numbers with this property.

I asked him to write down any number, no matter how large, and said I would immediately provide another number such that their sum and product would be the same. He liked the number 987654321, and I quickly wrote down the second number.

What exactly was it?

It turns out that for any predetermined number, there exists another number that, together with it, possesses this characteristic. If the reader is unaware of this, perhaps this problem will interest him, and he will try to find the corresponding pattern himself.
174. Taking any number, and then another, equal to 1 plus a fraction, where the numerator is 1 and the denominator is the number one less than the given one, we get a pair of numbers that give the same result when added and when multiplied. Here are a few examples: 3 and \(1 \frac{1}{2}, 4\) and \(1 \frac{1}{3}\), 5 and \(1 \frac{1}{4}\), etc. Therefore, upon receiving 987654321, I immediately wrote \(1 \frac{1}{987654320}\). The sum and the product in this case are \(987654322 \frac{1}{987654320}\).

The pair 2 and 2 is considered an exception because the denominator in this case is 1, and the second number also turns out to be an integer \(1 \frac{1}{1}=2\). However, it can be noticed that this case also follows the general rule. The number can be either an integer or a fraction, and the condition does not specify that we must find an integer, as in that case the only solution would indeed be the case of 2 and 2. Of course, decimal fractions are also allowed, such as 6 and 1.2; 11 and 1.1; 26 and 1.04.

Thus, the number corresponding to \(n\), paired with \(n\), has the form

\[
\frac{n}{n-1}=(n+1)+\frac{1}{n-1}
\]