[ Fundamental Theorem of Arithmetic. Prime Factorization]

Author: Blinkov A. d:

The product of five different integers is 2022. What can their sum be? If there are multiple answers, list them all.

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Let's factorize 2022 into prime factors: \(2022 = 2 \cdot 3 \cdot 337\), which means that out of the five multiplied numbers, only three can be greater than 1 in modulus. The remaining two must be equal to 1 in modulus; there are exactly two such numbers (1 and \(-1\)), so 2022 is obtained as \((-1) \cdot 1 \cdot (\pm 2) \cdot (\pm 3) \cdot (\pm 337)\). It remains to note that the minus sign must either stand before one of the numbers \(2, 3, 337\), or before all three - in total, there are 4 options and, accordingly, 4 possible sums: 338, 336, -332, -342.

## Answer

The problem