4. The sum of several natural numbers is 972. What is the greatest possible value of their product?
Solution. Let's consider into which addends 972 should be divided to obtain the maximum possible product. Among the addends, there should be no ones - if there is a one, we can add it to any of the other addends, and the product will increase. Any addend greater than 4 can be broken down into twos and threes. That is, if there is an addend $p$, we can subtract 2 or 3 from it, and add this two or three to the set. The product will increase, since $2 \cdot (p-2) > p, 3 \cdot (p-3) > p$. A possible addend 4 can also be replaced by two twos. Thus, to achieve the maximum possible product, the numbers that sum to 972 should be only 2 and 3. Further, any three twos are replaced by two threes, $2 \cdot 2 \cdot 2 < 3 \cdot 3$. Therefore, there can be only one, two, or no twos in the set. But if there is one or two twos, and the rest of the addends are threes, the entire sum will not be divisible by 3. The given number is divisible by 3. Therefore, to achieve the maximum possible product, this number should be represented as the sum of threes. $972 : 3 = 324$.

Answer. $3^{324}$.