# Task 8.2

Find the greatest value that the product of natural numbers can take, the sum of which is 2020.

## Number of points 7

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# Answer:

$2^{2} \cdot 3^{672}$

## Solution

If the product contains a number $x$ greater than 4, then by replacing it with the sum $3+(x-3)$, we get a larger product, since for such values of $x<3(3-x)$. If there is a term equal to 1, then combining it with any other term will yield a larger product. Therefore, the number 2020 should be represented as a sum of twos and threes.

We have

$2+2+2=3+3$, but $2 \cdot 2 \cdot 2<3 \cdot 3$

Therefore, the number of threes should be maximized. Thus, $2020=672 \cdot 3+2 \cdot 2$ and the maximum product is $2^{2} \cdot 3^{672}$

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