[^0]A company of several friends communicated in such a way that each email was received by everyone except the sender.

Each person wrote the same number of emails, resulting in a total of 440 emails received by everyone.

How many people could have been in this company?

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Let there be $n$ people in the company, and each sent $k$ letters. Then, from one person to the others, $k(n-1)$ letters were received, and from all the senders, $k(n-1)n$ letters were sent. Therefore, the number 440 is the product of three factors, two of which differ by 1.

Since $440=2^{3} \cdot 5 \cdot 11$, we can proceed with the enumeration under the following constraints:

1) $n < 440$;
2) the numbers $n$ and $n-1$ do not contain any prime factors other than 2, 5, and 11.

As a result, we get three options: $440=2 \cdot 10 \cdot 11=22 \cdot 4 \cdot 5=220 \cdot 1 \cdot 2$.

## Answer

2, 5 or 11.