8. (5 points) The founder of a noble family received a plot of land. Each man in the family, upon dying, divided the land he inherited equally among his sons. If he had no sons, the land went to the state. No other members of the family gained or lost any land in any other way. In total, there were 150 people in the family. What is the smallest fraction of the original plot of land that any member of the family could have received?
Answer: $\frac{1}{2 \cdot 3^{49}}$

Solution: Consider the person with the smallest share. Let his father have $a_{1}$ sons, his grandfather $a_{2}$, and so on up to the founder of the lineage, who had $a_{n}$. Then the share of this person is $\frac{1}{a_{1} a_{2} \ldots a_{n}}$, and we know that $a_{1}+a_{2}+\ldots+a_{n}$ does not exceed 149 (since the founder of the lineage is definitely not included in this sum).

Thus, we are looking for a set of numbers with a sum not exceeding 149 and the maximum product. We will prove that any set, except for the set consisting of 49 threes and one two, does not have the maximum product, as it can be increased.

First, if the sum of the numbers in the set is less than 149, add another number to make it equal to 149. The product will not decrease.

Second, suppose there is a number $a \geqslant 4$ in our set. Replace $a$ with a pair of numbers $b$ and $c$ greater than one, such that $b+c=a$. We will prove that $b c \geqslant b+c$. Indeed, $b c-b-c=(b-1)(c-1)-1$, which is non-negative for $b$ and $c$ greater than one.

Third, if there is a number 1 in the set, then we can replace any number $a$ and 1 with $a+1$, thereby increasing the product of these numbers by 1.

Thus, any set can be transformed into a set of twos and threes without decreasing the product of its numbers.

Furthermore, if there are at least three twos, their product is 8. If we replace three twos with two threes, the sum of the numbers remains the same, but the product becomes 9, which is again increased. Thus, we can ensure that the number of twos is no more than two.

Since $149=49 \cdot 3+2$, the final set will have exactly one two.

Therefore, any set of natural numbers with a sum not exceeding 149 can be transformed into a set of 49 threes and one two, and the product of the numbers in the set will not decrease. This means that the final set has the largest product, equal to $3^{49} \cdot 2$, which gives us the smallest inheritance share as the reciprocal of this number.