4. The number 2090 is written as the product of six different integers. Determine the smallest value of the sum of these numbers.
4. The number 2090 is written as the product of six different integers. Determine the smallest value of the sum of these numbers.

## Solution.

Let's factorize the number 2090 into prime factors:

$$
\begin{aligned}
2090 & =209 \cdot 10 \\
& =11 \cdot 19 \cdot 2 \cdot 5
\end{aligned}
$$

1 POINT

If the number is written as the product of six different integers and it has four prime factors, then two of the factors of this product must be the numbers 1 and -1.

2 POINTS

Since the number is positive and we want the smallest value of the sum, we need to have an even number of negative factors, and the factors with the largest absolute values should be negative.
$2090=(-19) \cdot(-11) \cdot(-5) \cdot 2 \cdot(-1) \cdot 1$
2 POINTS
The smallest value of the sum is $-19+(-11)+(-5)+2+(-1)+1=-33$.
1 POINT

TOTAL 6 POINTS